Unit Circle Guided Notes π
Unit Circle Guided Notes
Interactive Slide Companion • Foundations, Coordinates & Angle Rotations
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Part 1: Why Radius = 1? (Slide 2) Equation: \(x^2 + y^2 = \_\_\_\_\)
Cosine \(\rightarrow\) Horizontal
\(\cos\theta = \frac{\text{adjacent}}{\text{hyp}} = \frac{x}{\_\_\_\_} = \mathbf{\_\_\_\_}\)
Point x-coordinate is \(\cos\theta\)
Sine \(\rightarrow\) Vertical
\(\sin\theta = \frac{\text{opposite}}{\text{hyp}} = \frac{y}{\_\_\_\_} = \mathbf{\_\_\_\_}\)
Point y-coordinate is \(\sin\theta\)
Tangent \(\rightarrow\) Slope / Ratio
\(\tan\theta = \frac{\text{opp}}{\text{adj}} = \frac{\_\_\_\_}{\_\_\_\_} = \frac{y}{x}\)
Undefined when \(x = \_\_\_\_\)
Degrees to Radians Blueprint (Slide 3)
Degrees \(\rightarrow\) Radians: \(\text{Rad} = \text{Deg} \times \frac{\pi}{\_\_\_\_\_\_}\)
Radians \(\rightarrow\) Degrees: \(\text{Deg} = \text{Rad} \times \frac{\_\_\_\_\_\_}{\pi}\)
\(/6\) \(\_\_\_\_^\circ\) family
\(/4\) \(\_\_\_\_^\circ\) family
\(/3\) \(\_\_\_\_^\circ\) family
\(/2\) Quadrantal
Coterminal & CFU 1 (Slides 4–5) \(\theta \pm 360^\circ k\)
Share the exact same terminal ray. Standard range: \([0, 360^\circ)\) / \([0, 2\pi)\).
1. Convert \(240^\circ\): ____________
2. Coterminal to \(-\frac{2\pi}{3}\): ____________
3. Quadrant of \(495^\circ\): Quadrant _____
Special Triangles (\(r=1\)) (Slide 6) Hypotenuse \(= 1\)
\(30^\circ - 60^\circ - 90^\circ\)
Short leg (opp \(30^\circ\)) = ______
Long leg (opp \(60^\circ\)) = ______
\(45^\circ - 45^\circ - 90^\circ\)
Both equal legs =
______________
ASTC Sign Rule (Slide 7) All Students Take Calculus
QII: S Only ______ (+) \(\cos, \tan < 0\)
QI: A ______ are (+) \(\sin, \cos, \tan > 0\)
QIII: T Only ______ (+) \(\sin, \cos < 0\)
QIV: C Only ______ (+) \(\sin, \tan < 0\)
The Completed Unit Circle Map (Slide 8 Companion)
Fill in missing radians, degrees, and coordinate pairs \((x, y)\)
Quadrant II \((-, +)\)
\(120^\circ\) (\(2\pi/3\)): \((-\frac{1}{2}, \_\_\_\_)\)
\(135^\circ\) (\(3\pi/4\)): \((\_\_\_\_, \frac{\sqrt{2}}{2})\)
\(150^\circ\) (\(5\pi/6\)): \((-\frac{\sqrt{3}}{2}, \_\_\_\_)\)
Quadrant III \((-, -)\)
\(210^\circ\) (\(7\pi/6\)): \((-\frac{\sqrt{3}}{2}, \_\_\_\_)\)
\(225^\circ\) (\(5\pi/4\)): \((\_\_\_\_, -\frac{\sqrt{2}}{2})\)
\(240^\circ\) (\(4\pi/3\)): \((-\frac{1}{2}, \_\_\_\_)\)
x y (1, 0) (0, 1) (-1, 0) (0, -1) QI QII QIII QIV (0,0)
Quadrant I \((+, +)\)
\(30^\circ\) (\(\pi/6\)): \((\frac{\sqrt{3}}{2}, \_\_\_\_)\)
\(45^\circ\) (\(\pi/4\)): \((\_\_\_\_, \frac{\sqrt{2}}{2})\)
\(60^\circ\) (\(\pi/3\)): \((\frac{1}{2}, \_\_\_\_)\)
Quadrant IV \((+, -)\)
\(300^\circ\) (\(5\pi/3\)): \((\frac{1}{2}, \_\_\_\_)\)
\(315^\circ\) (\(7\pi/4\)): \((\_\_\_\_, -\frac{\sqrt{2}}{2})\)
\(330^\circ\) (\(11\pi/6\)): \((\frac{\sqrt{3}}{2}, \_\_\_\_)\)
Unit Circle Guided Notes • Aligned with Lecture Presentation Page 1 of 2
θ′
Reference Angles & Exact Value Protocol
Part 2: Reduction Formulas, Understanding Checks & Scaffolded Practice
Slides 9–12 Companion
Reference Angle Rule (\(\theta'\)): Acute distance to the x-axis (Slide 9)
Never measure to y-axis! Range: \(0 < \theta' \le 90^\circ\)
Quadrant I
Acute standard angle
\(\theta' = \mathbf{\_\_\_\_}\)
Rad: \(\theta' = \theta\)
Quadrant II
Distance to \(180^\circ\)
\(\theta' = 180^\circ - \mathbf{\_\_\_\_}\)
Rad: \(\pi - \mathbf{\_\_\_\_}\)
Quadrant III
Distance past \(180^\circ\)
\(\theta' = \mathbf{\_\_\_\_} - 180^\circ\)
Rad: \(\mathbf{\_\_\_\_} - \pi\)
Quadrant IV
Distance to \(360^\circ\)
\(\theta' = 360^\circ - \mathbf{\_\_\_\_}\)
Rad: \(2\pi - \mathbf{\_\_\_\_}\)
Check for Understanding 2: Active Recall (Slide 10) Whiteboard Prompts
\(\bullet\) Ref angle of \(\frac{7\pi}{4}\): ______
\(\bullet\) Sign of \(\tan\left(\frac{5\pi}{6}\right)\): ______
\(\bullet\) \(\sec(210^\circ)\) is (+/-) because: __________
The 3-Step Exact Value Protocol (Slide 11) \(\text{Trig}(\theta) = \pm\,\text{Trig}(\theta')\)
1
Quadrant & Sign: Determine quadrant of \(\theta\) and sign \((+/-)\) using ASTC.
2
Reference Angle: Calculate acute distance \(\theta'\) (\(30^\circ, 45^\circ, 60^\circ\)) to the horizontal x-axis.
3
Evaluate QI & Attach Sign: Find QI value for \(\theta'\) and combine: \(\pm\,\text{Trig}(\theta')\).
Lecture Model: Evaluate \(\cos\left(\frac{5\pi}{6}\right)\)
1. Quad: \(\frac{5\pi}{6} \in \text{Q\_\_\_} \implies \cos\) is ______
2. Ref Angle: \(\theta' = \pi - \frac{5\pi}{6} = \mathbf{\_\_\_\_\_\_}\)
3. QI Value: \(\cos\left(\frac{\pi}{6}\right) = \mathbf{\_\_\_\_\_\_}\)
Final Answer: \(\cos\left(\frac{5\pi}{6}\right) = \mathbf{\_\_\_\_\_\_}\)
CFU 3 Exit Showdown & Error Analysis (Slide 12) Critical Check
A. \(\sin\left(\frac{4\pi}{3}\right)\): QIII \(\implies\) Sign: ____, \(\theta' = \mathbf{\_\_\_\_}\), Ans: ______
B. \(\cos\left(-\frac{\pi}{4}\right)\): QIV \(\implies\) Sign: ____, \(\theta' = \mathbf{\_\_\_\_}\), Ans: ______
Spot the Error: Student claimed \(\tan\left(\frac{2\pi}{3}\right) = \sqrt{3}\)
What did they forget? __________________________________________
Correct value: \(\tan\left(\frac{2\pi}{3}\right) = \mathbf{\_\_\_\_\_\_\_\_}\)
Guided Application Practice: Evaluate Each Exact Value
State Quadrant, Sign, Reference Angle (\(\theta'\)), and Solution
1. \(\sin(150^\circ)\) Degrees
Quad: ____
Sign: ____
\(\theta'\): ____
Answer: ____________
2. \(\cos(225^\circ)\) Degrees
Quad: ____
Sign: ____
\(\theta'\): ____
Answer: ____________
3. \(\tan\left(\frac{4\pi}{3}\right)\) Radians
Quad: ____
Sign: ____
\(\theta'\): ____
Answer: ____________
4. \(\cos(330^\circ)\) Degrees
Quad: ____
Sign: ____
\(\theta'\): ____
Answer: ____________
5. \(\sin\left(-\frac{2\pi}{3}\right)\) Negative Rad
Coterm: __
Quad: ____
\(\theta'\): ____
Answer: ____________
6. \(\sec(120^\circ)\) Reciprocal
Quad: ____
\(\cos(120^\circ)\):
\(\text{Flip}\): ____
Answer: ____________
Unit Circle Guided Notes • Precalculus / Trigonometry Page 2 of 2
Circle Values Practice Worksheet Circle Values Practice Worksheet
Unit Circle Rotations, Reference Angles & Exact Evaluations
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1 Coterminal Angles & Quadrant Location
Find a coterminal angle in \([0, 360^\circ)\) or \([0, 2\pi)\) and name the quadrant.
1. \(\theta = 420^\circ\)
Coterminal: ________
Quadrant: ________
2. \(\theta = -135^\circ\)
Coterminal: ________
Quadrant: ________
3. \(\theta = \frac{11\pi}{4}\)
Coterminal: ________
Quadrant: ________
4. \(\theta = -\frac{7\pi}{6}\)
Coterminal: ________
Quadrant: ________
2 Reference Angles (\(\theta'\)) & Function Signs
Determine the acute reference angle and sign (\(+\) or \(-\)) via ASTC.
5. \(\theta = 225^\circ\)
\(\theta'\): ____________
Sign of \(\tan\theta\): _____
6. \(\theta = 330^\circ\)
\(\theta'\): ____________
Sign of \(\cos\theta\): _____
7. \(\theta = \frac{5\pi}{3}\)
\(\theta'\): ____________
Sign of \(\sin\theta\): _____
8. \(\theta = \frac{3\pi}{4}\)
\(\theta'\): ____________
Sign of \(\cos\theta\): _____
3 Exact Trigonometric Values (Primary: \(\sin, \cos, \tan\))
No decimals. Write exact simplified radicals or fractions.
9. \(\sin(210^\circ)\)
Quad / \(\theta'\): _________
Ans: ________
10. \(\cos(135^\circ)\)
Quad / \(\theta'\): _________
Ans: ________
11. \(\tan(240^\circ)\)
Quad / \(\theta'\): _________
Ans: ________
12. \(\cos(180^\circ)\)
Quadrantal Axis
Ans: ________
13. \(\sin\left(\frac{5\pi}{6}\right)\)
Quad / \(\theta'\): _________
Ans: ________
14. \(\cos\left(\frac{7\pi}{4}\right)\)
Quad / \(\theta'\): _________
Ans: ________
15. \(\tan\left(\frac{4\pi}{3}\right)\)
Quad / \(\theta'\): _________
Ans: ________
16. \(\sin\left(\frac{3\pi}{2}\right)\)
Quadrantal Axis
Ans: ________
Circle Values Practice Worksheet • Unit Circle Mastery Page 1 of 2
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Reciprocals, Expressions & Modeling Applications
Advanced Trigonometric Practice
4 Reciprocal Functions (\(\sec\theta = \frac{1}{\cos}, \csc\theta = \frac{1}{\sin}, \cot\theta = \frac{1}{\tan}\))
Rationalize denominators when needed.
17. \(\csc(30^\circ)\) \(1/\sin\)
\(\sin(30^\circ) = \) ____________
Value: ________
18. \(\sec\left(\frac{2\pi}{3}\right)\) \(1/\cos\)
\(\cos\left(\frac{2\pi}{3}\right) = \) ____________
Value: ________
19. \(\cot(225^\circ)\) \(1/\tan\)
\(\tan(225^\circ) = \) ____________
Value: ________
20. \(\sec\left(\frac{11\pi}{6}\right)\) \(1/\cos\)
\(\cos\left(\frac{11\pi}{6}\right) = \) ____________
Value: ________
21. \(\csc\left(-\frac{\pi}{3}\right)\) Negative Angle
\(\sin\left(-\frac{\pi}{3}\right) = \) ____________
Value: ________
22. \(\cot\left(\frac{\pi}{2}\right)\) \(\cos/\sin\)
\((x, y) = (0, 1)\)
Value: ________
5 Evaluating Combined Expressions
Substitute exact values, simplify fractions, and combine terms.
23. Evaluate: \(2\cos(60^\circ) + 4\sin(30^\circ)\)
Substitute exact values for \(\cos(60^\circ)\) and \(\sin(30^\circ)\):
Final Value: ________
24. Evaluate: \(\sin^2\left(\frac{\pi}{4}\right) + \cos^2\left(\frac{\pi}{4}\right)\)
Square each coordinate value:
Final Value: ________
25. Evaluate: \(\tan\left(\frac{\pi}{3}\right) \cdot \cot\left(\frac{\pi}{3}\right) - \cos(\pi)\)
Evaluate products and observe signs carefully:
Final Value: ________
26. Evaluate: \(\frac{\sin(90^\circ) + \cos(0^\circ)}{\tan(45^\circ)}\)
Evaluate numerator and denominator separately:
Final Value: ________
6 Coordinate Geometry & Angle Reconstruction
Show your mathematical reasoning in the box provided.
27. Unit Circle Coordinate Search
Point \(P\left(-\frac{1}{2}, y\right)\) lies on the unit circle in Quadrant II. Find \(y > 0\), and write the exact values of \(\sin\theta\) and \(\tan\theta\).
\(y =\) ________ | \(\tan\theta =\) ________
28. Reconstructing the Angle
Given that \(\tan\theta = -\sqrt{3}\) and \(\sin\theta > 0\), determine the quadrant, the exact angle \(\theta \in [0, 2\pi)\), and find \(\cos\theta\).
\(\theta =\) ________ | \(\cos\theta =\) ________
Circle Values Practice Worksheet • Unit Circle Mastery Page 2 of 2
Circle Values Answer Key Circle Values Answer Key
Teacher Solutions
Full Step-by-Step Solutions & Grading Breakdown • Page 1 of 2
Unit Circle Mastery
Practice Worksheet Key
1 Part 1 Solutions: Coterminal Angles & Quadrants
Rule: \(\theta \pm 360^\circ\) or \(\theta \pm 2\pi\)
1. \(\theta = 420^\circ\)
\(420^\circ - 360^\circ = 60^\circ\)
Coterminal: \(60^\circ\)
Quadrant: Quadrant I
2. \(\theta = -135^\circ\)
\(-135^\circ + 360^\circ = 225^\circ\)
Coterminal: \(225^\circ\)
Quadrant: Quadrant III
3. \(\theta = \frac{11\pi}{4}\)
\(\frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4}\)
Coterminal: \(\frac{3\pi}{4}\)
Quadrant: Quadrant II
4. \(\theta = -\frac{7\pi}{6}\)
\(-\frac{7\pi}{6} + \frac{12\pi}{6} = \frac{5\pi}{6}\)
Coterminal: \(\frac{5\pi}{6}\)
Quadrant: Quadrant II
2 Part 2 Solutions: Reference Angles (\(\theta'\)) & Signs
ASTC: All(+) in QI, Sin(+) in QII, Tan(+) in QIII, Cos(+) in QIV
5. \(\theta = 225^\circ\) (QIII)
\(225^\circ - 180^\circ = 45^\circ\)
\(\theta'\): \(45^\circ\)
Sign of \(\tan\theta\): Positive (+)
6. \(\theta = 330^\circ\) (QIV)
\(360^\circ - 330^\circ = 30^\circ\)
\(\theta'\): \(30^\circ\)
Sign of \(\cos\theta\): Positive (+)
7. \(\theta = \frac{5\pi}{3}\) (QIV)
\(2\pi - \frac{5\pi}{3} = \frac{\pi}{3}\)
\(\theta'\): \(\frac{\pi}{3}\)
Sign of \(\sin\theta\): Negative (−)
8. \(\theta = \frac{3\pi}{4}\) (QII)
\(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\)
\(\theta'\): \(\frac{\pi}{4}\)
Sign of \(\cos\theta\): Negative (−)
3 Part 3 Solutions: Exact Values (Primary Trig Functions)
Work: [Quadrant] → [Sign] → [\(\theta'\) value]
9. \(\sin(210^\circ)\)
QIII: \((-)\), \(\theta' = 30^\circ\)
\(-\sin(30^\circ) = -\frac{1}{2}\)
Answer: \(-\frac{1}{2}\)
10. \(\cos(135^\circ)\)
QII: \((-)\), \(\theta' = 45^\circ\)
\(-\cos(45^\circ) = -\frac{\sqrt{2}}{2}\)
Answer: \(-\frac{\sqrt{2}}{2}\)
11. \(\tan(240^\circ)\)
QIII: \((+)\), \(\theta' = 60^\circ\)
\(+\tan(60^\circ) = +\sqrt{3}\)
Answer: \(\sqrt{3}\)
12. \(\cos(180^\circ)\)
Axis: \((-1, 0)\)
\(x\)-coord at \(180^\circ = -1\)
Answer: \(-1\)
13. \(\sin\left(\frac{5\pi}{6}\right)\)
QII: \((+)\), \(\theta' = \frac{\pi}{6}\)
\(+\sin\left(\frac{\pi}{6}\right) = +\frac{1}{2}\)
Answer: \(\frac{1}{2}\)
14. \(\cos\left(\frac{7\pi}{4}\right)\)
QIV: \((+)\), \(\theta' = \frac{\pi}{4}\)
\(+\cos\left(\frac{\pi}{4}\right) = +\frac{\sqrt{2}}{2}\)
Answer: \(\frac{\sqrt{2}}{2}\)
15. \(\tan\left(\frac{4\pi}{3}\right)\)
QIII: \((+)\), \(\theta' = \frac{\pi}{3}\)
\(+\tan\left(\frac{\pi}{3}\right) = +\sqrt{3}\)
Answer: \(\sqrt{3}\)
16. \(\sin\left(\frac{3\pi}{2}\right)\)
Axis: \((0, -1)\)
\(y\)-coord at \(\frac{3\pi}{2} = -1\)
Answer: \(-1\)
Circle Values Answer Key • Teacher Resource Page 1 of 2
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Part 2 Solutions: Reciprocals, Combined Expressions & Modeling
Page 2 of 2
4 Part 4 Solutions: Reciprocal Functions
Verify inverted fractions and rationalized radicals.
17. \(\csc(30^\circ)\)
\(\sin(30^\circ) = \frac{1}{2}\)
\(\csc = \frac{1}{\sin} = \frac{1}{1/2} = 2\)
Value: \(2\)
18. \(\sec\left(\frac{2\pi}{3}\right)\)
\(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\) (QII)
\(\sec = \frac{1}{\cos} = \frac{1}{-1/2} = -2\)
Value: \(-2\)
19. \(\cot(225^\circ)\)
\(\tan(225^\circ) = +1\) (QIII)
\(\cot = \frac{1}{\tan} = \frac{1}{1} = 1\)
Value: \(1\)
20. \(\sec\left(\frac{11\pi}{6}\right)\)
\(\cos\left(\frac{11\pi}{6}\right) = +\frac{\sqrt{3}}{2}\) (QIV)
\(\sec = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\)
Value: \(\frac{2\sqrt{3}}{3}\)
21. \(\csc\left(-\frac{\pi}{3}\right)\)
\(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\) (QIV)
\(\csc = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\)
Value: \(-\frac{2\sqrt{3}}{3}\)
22. \(\cot\left(\frac{\pi}{2}\right)\)
Point \((x, y) = (0, 1)\)
\(\cot = \frac{x}{y} = \frac{0}{1} = 0\)
Value: \(0\)
5 Part 5 Solutions: Combined Expressions
Shows exact substitution and simplification steps.
23. \(2\cos(60^\circ) + 4\sin(30^\circ)\)
\(= 2\left(\frac{1}{2}\right) + 4\left(\frac{1}{2}\right) = 1 + 2 = 3\)
Final Value: \(3\)
24. \(\sin^2\left(\frac{\pi}{4}\right) + \cos^2\left(\frac{\pi}{4}\right)\)
\(= \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} + \frac{2}{4} = \frac{4}{4} = 1\)
Final Value: \(1\) (Pythagorean Identity)
25. \(\tan\left(\frac{\pi}{3}\right) \cdot \cot\left(\frac{\pi}{3}\right) - \cos(\pi)\)
\(= (\sqrt{3})\left(\frac{1}{\sqrt{3}}\right) - (-1) = 1 - (-1) = 2\)
Final Value: \(2\)
26. \(\frac{\sin(90^\circ) + \cos(0^\circ)}{\tan(45^\circ)}\)
\(= \frac{1 + 1}{1} = \frac{2}{1} = 2\)
Final Value: \(2\)
6 Part 6 Solutions: Coordinate Modeling
27. Unit Circle Coordinate Search: \(P\left(-\frac{1}{2}, y\right)\) in QII
\(x^2 + y^2 = 1 \implies \left(-\frac{1}{2}\right)^2 + y^2 = 1 \implies \frac{1}{4} + y^2 = 1 \implies y^2 = \frac{3}{4}\).
\(y = \frac{\sqrt{3}}{2}\) | \(\tan\theta = -\sqrt{3}\)
28. Angle Reconstruction: \(\tan\theta = -\sqrt{3}\), \(\sin\theta > 0\)
\(\tan < 0\) in QII & QIV. \(\sin > 0\) in QI & QII. Both hold \(\implies \theta \in \text{QII}\).
\(\theta = \frac{2\pi}{3}\) (\(120^\circ\)) | \(\cos\theta = -\frac{1}{2}\)
Common Student Misconceptions & Scoring Tips
y-Axis Reference Error: Watch for students measuring reference angles to \(90^\circ\) or \(270^\circ\) (e.g. claiming the ref angle for \(150^\circ\) is \(60^\circ\) instead of \(30^\circ\)).
Reciprocal Inversion: Remind students that \(\sec\theta = 1/\cos\theta\) (starts with 's' and 'c'), whereas \(\csc\theta = 1/\sin\theta\) ('c' and 's'). Never invert the sign when taking the reciprocal!
Suggested Grading (32 pts): Part 1: 4 pts (1 pt each). Part 2: 4 pts. Part 3: 8 pts. Part 4: 6 pts. Part 5: 4 pts. Part 6: 6 pts (3 pts each). Partial credit for correct reference angle.
Circle Values Answer Key • Unit Circle Mastery Page 2 of 2
Unit Circle Presentation Slides π
Precalculus & Trigonometry
Unit Circle Mastery
The Unit Circle Navigator
Connecting Angles, Coordinates, and the Fundamental Geometry of Trigonometric Functions
Radian Rotations Coordinate Pairs \((x, y)\) Active Understanding Checks
Foundation Principle
Why Radius = 1?
\(x^2 + y^2 = 1\)
Horizontal Axis
\(x = \cos\theta\)
Because hypotenuse \(r = 1\), the adjacent leg simplifies directly to the x-coordinate:
\(\frac{\text{adjacent}}{1} = x\)
Vertical Axis
\(y = \sin\theta\)
The opposite leg directly represents the vertical coordinate of the terminal point:
\(\frac{\text{opposite}}{1} = y\)
Slope / Ratio
\(\tan\theta = \frac{y}{x}\)
Tangent is the slope of the terminal ray: rise over run, undefined when \(x = 0\):
\(\frac{\sin\theta}{\cos\theta} = \frac{y}{x}\)
Every point on the unit circle is explicitly \(P(\cos\theta, \sin\theta)\) Slide 2 / 12
Rotational Language
Degrees to Radians Blueprint
\(180^\circ = \pi \text{ radians}\)
Conversion Bridges
Degrees \(\rightarrow\) Radians:
\(\text{Rad} = \text{Deg} \times \frac{\pi}{180^\circ}\)
Radians \(\rightarrow\) Degrees:
\(\text{Deg} = \text{Rad} \times \frac{180^\circ}{\pi}\)
Radian Denominator Patterns
Denominator \(/6\): \(30^\circ\) family (\(\pi/6, 5\pi/6\))
Denominator \(/4\): \(45^\circ\) family (\(\pi/4, 3\pi/4\))
Denominator \(/3\): \(60^\circ\) family (\(\pi/3, 2\pi/3\))
Denominator \(/2\): Quadrantal Axes (\(\pi/2, 3\pi/2\))
Mastering radian denominators removes the need for brute-force memorization Slide 3 / 12
Rotational Equivalence
Coterminal & Negative Angles
\(\theta \pm 360^\circ k\) | \(\theta \pm 2\pi k\)
What Are Coterminal Angles?
Angles in standard position that share the exact same terminal ray, and therefore produce identical coordinate pairs \((x, y)\).
Positive Rotations: Counter-clockwise from the positive x-axis.
Negative Rotations: Clockwise from the positive x-axis (add \(360^\circ\) or \(2\pi\)).
Standard Range: \([0, 360^\circ)\) & \([0, 2\pi)\)
Angle > \(360^\circ\): \(\theta = 420^\circ \rightarrow 420^\circ - 360^\circ\)
\(60^\circ\) (QI)
Negative Degrees: \(\theta = -135^\circ \rightarrow -135^\circ + 360^\circ\)
\(225^\circ\) (QIII)
Negative Radians: \(\theta = -\frac{\pi}{4} \rightarrow -\frac{\pi}{4} + \frac{8\pi}{4}\)
\(\frac{7\pi}{4}\) (QIV)
Trigonometric values are strictly periodic: \(\sin(\theta \pm 2\pi) = \sin(\theta)\) and \(\cos(\theta \pm 2\pi) = \cos(\theta)\) Slide 4 / 12
Formative Check 1 • Think-Pair-Share
Check for Understanding: Rotations & Radians
Pause & Discuss (2 Min)
1 Degree \(\rightarrow\) Radian
Convert \(240^\circ\) to exact radian measure in terms of \(\pi\).
Solution & Strategy: \(240^\circ \times \frac{\pi}{180^\circ} = \frac{4\pi}{3}\) Belongs to the \(\pi/3\) family in QIII
2 Coterminal Target
Find the positive angle in \([0, 2\pi)\) coterminal to \(-\frac{2\pi}{3}\).
Solution & Strategy: \(-\frac{2\pi}{3} + \frac{6\pi}{3} = \frac{4\pi}{3}\) Add full \(2\pi\) rotation to enter standard range
3 Quadrant Hunter
Which quadrant does \(\theta = 495^\circ\) terminate in?
Solution & Strategy: \(495^\circ - 360^\circ = 135^\circ\) \(\implies \text{Quadrant II}\) \((90^\circ < 135^\circ < 180^\circ)\)
Use whiteboard responses or hand signals (1, 2, 3, 4 for quadrants) before revealing Slide 5 / 12
Geometric Derivations
Special Right Triangles (\(r = 1\))
Hypotenuse = 1
\(30^\circ - 60^\circ - 90^\circ\) Angles: \(\pi/6, \pi/3\)
Sides scale from the standard \(1 - \sqrt{3} - 2\) triangle by dividing by 2:
Short Leg (opp \(30^\circ\))
\(\frac{1}{2} = 0.5\)
Long Leg (opp \(60^\circ\))
\(\frac{\sqrt{3}}{2} \approx 0.866\)
\(\bullet\) At \(30^\circ\): \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\) | \(\bullet\) At \(60^\circ\): \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
\(45^\circ - 45^\circ - 90^\circ\) Angle: \(\pi/4\)
Sides scale from \(1 - 1 - \sqrt{2}\) by dividing each leg by \(\sqrt{2}\):
Equal Symmetrical Legs:
\(\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.707\)
Coordinate Pair at \(45^\circ\): \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)
Every coordinate on the unit circle is formed by reflecting these two triangles Slide 6 / 12
Sign Conventions
ASTC Quadrant Rule
All Students Take Calculus
A
Quadrant I
\((x > 0, y > 0)\)
ALL Positive \(\sin, \cos, \tan > 0\)
\(0^\circ\) to \(90^\circ\)
S
Quadrant II
\((x < 0, y > 0)\)
SINE Positive \(\cos < 0, \tan < 0\)
\(90^\circ\) to \(180^\circ\)
T
Quadrant III
\((x < 0, y < 0)\)
TANGENT Positive \(\sin < 0, \cos < 0\)
\(180^\circ\) to \(270^\circ\)
C
Quadrant IV
\((x > 0, y < 0)\)
COSINE Positive \(\sin < 0, \tan < 0\)
\(270^\circ\) to \(360^\circ\)
Reciprocal functions (\(\csc, \sec, \cot\)) always inherit the exact sign of their primary function Slide 7 / 12
Master Coordinate Map
The Completed Unit Circle
\(P(x, y) = (\cos\theta, \sin\theta)\)
Quadrant II \((-, +)\) \(\sin > 0\)
\(120^\circ\) \(\left(\frac{2\pi}{3}\right)\) \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
\(135^\circ\) \(\left(\frac{3\pi}{4}\right)\) \(\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)
\(150^\circ\) \(\left(\frac{5\pi}{6}\right)\) \(\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
Quadrant III \((-, -)\) \(\tan > 0\)
\(210^\circ\) \(\left(\frac{7\pi}{6}\right)\) \(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)
\(225^\circ\) \(\left(\frac{5\pi}{4}\right)\) \(\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\)
\(240^\circ\) \(\left(\frac{4\pi}{3}\right)\) \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
x y QI QII QIII QIV (1, 0) (0, 1) (-1, 0) (0, -1)
\(30^\circ / \frac{\pi}{6}\)
\(45^\circ / \frac{\pi}{4}\)
\(60^\circ / \frac{\pi}{3}\)
Quadrant I \((+, +)\) \(\text{ALL} > 0\)
\(30^\circ\) \(\left(\frac{\pi}{6}\right)\) \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
\(45^\circ\) \(\left(\frac{\pi}{4}\right)\) \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)
\(60^\circ\) \(\left(\frac{\pi}{3}\right)\) \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
Quadrant IV \((+, -)\) \(\cos > 0\)
\(300^\circ\) \(\left(\frac{5\pi}{3}\right)\) \(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
\(315^\circ\) \(\left(\frac{7\pi}{4}\right)\) \(\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\)
\(330^\circ\) \(\left(\frac{11\pi}{6}\right)\) \(\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)
Notice the symmetry: all 16 coordinates stem from reflections of the Quadrant I special triangles Slide 8 / 12
Reduction Formula
Reference Angles (\(\theta'\))
Golden Rule: Always measure to the x-axis!
Quadrant I
Acute Angle
\(\theta' = \theta\)
\(\theta' = \theta\)
Quadrant II
Distance to \(180^\circ\)
\(180^\circ - \theta\)
\(\pi - \theta\)
Quadrant III
Past \(180^\circ\)
\(\theta - 180^\circ\)
\(\theta - \pi\)
Quadrant IV
Distance to \(360^\circ\)
\(360^\circ - \theta\)
\(2\pi - \theta\)
Reference angles are strictly acute: \(0^\circ < \theta' \le 90^\circ\) or \(0 < \theta' \le \frac{\pi}{2}\) Slide 9 / 12
Formative Check 2 • Active Recall
Check for Understanding: Signs & Reference Angles
Whiteboard Showdown
1 Find Reference Angle
What is the reference angle \(\theta'\) for \(\theta = \frac{7\pi}{4}\)?
Solution & Strategy: \(\frac{7\pi}{4} \in \text{QIV} \implies \theta' = 2\pi - \frac{7\pi}{4}\) \(\theta' = \frac{\pi}{4}\) (\(45^\circ\))
2 ASTC Sign Predictor
Is \(\tan\left(\frac{5\pi}{6}\right)\) positive or negative? Explain using ASTC.
Solution & Strategy: \(\frac{5\pi}{6} \in \text{QII}\). In QII, only sine is positive. \(\implies \tan\left(\frac{5\pi}{6}\right)\) is Negative (−)
3 Spot the Trap
True or False: "Because \(210^\circ \in \text{QIII}\) where tangent is positive, \(\sec(210^\circ)\) is positive."
Correction & Rule: FALSE! \(\sec\theta = 1/\cos\theta\) In QIII, \(\cos < 0 \implies \sec(210^\circ)\) is Negative!
Reference angles are always positive distances measured strictly to the horizontal x-axis Slide 10 / 12
Master Protocol
Evaluating Exact Trig Values
3-Step Method
The 3 Essential Steps
1
Locate Quadrant & ASTC Sign
Determine whether the trig function is \(+\) or \(-\).
2
Find Reference Angle \(\theta'\)
Calculate acute distance to the horizontal x-axis.
3
Evaluate QI Value & Attach Sign
Apply: \(\text{Trig}(\theta) = \pm\,\text{Trig}(\theta')\).
Model Example
Evaluate \(\cos\left(\frac{5\pi}{6}\right)\)
1. Quadrant: \(\frac{5\pi}{6} \in \text{QII} \implies \cos\) is Negative (−)
2. Ref Angle: \(\theta' = \pi - \frac{5\pi}{6} = \frac{\pi}{6}\) (\(30^\circ\))
3. QI Value: \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)
Final Exact Value:
\(\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}\)
Mastering these 3 steps lets you evaluate any angle in less than 15 seconds Slide 11 / 12
Formative Check 3 • Exit Practice
Check for Understanding: Exact Trig Values
Final Mastery Check
Problem A: Evaluate \(\sin\left(\frac{4\pi}{3}\right)\) QIII
1. Sign: QIII \(\implies \sin\) is Negative (−) 2. \(\theta' = \frac{4\pi}{3} - \pi = \frac{\pi}{3}\)
\(\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)
Problem B: Evaluate \(\cos\left(-\frac{\pi}{4}\right)\) QIV
1. Sign: QIV \(\implies \cos\) is Positive (+) 2. \(\theta' = \frac{\pi}{4}\)
\(\cos\left(-\frac{\pi}{4}\right) = +\frac{\sqrt{2}}{2}\)
! Spot the Critical Mistake
Student Work: "Evaluate \(\tan\left(\frac{2\pi}{3}\right)\): Ref angle is \(\frac{\pi}{3}\). Since \(\tan(\pi/3) = \sqrt{3}\), my answer is \(\sqrt{3}\)."
What critical step did they skip?
They forgot Step 1: Check Quadrant Sign! \(\frac{2\pi}{3}\) terminates in Quadrant II, where tangent is negative.
Correct Value:
\(\tan\left(\frac{2\pi}{3}\right) = -\sqrt{3}\)
Now transition seamlessly to the Circle Values Practice Worksheet for independent application Slide 12 / 12
Unit Circle Mastery Quiz Q
Unit Circle Mastery Quiz
Form A
Radians, Reference Angles & Exact Trigonometric Values
Name:
Date:
Score: / 25
A Concept Checks (1 pt each)
Circle the correct letter.
1. What is the reference angle \(\theta'\) for \(\theta = 150^\circ\)?
A) \(30^\circ\)
B) \(60^\circ\)
C) \(150^\circ\)
D) \(45^\circ\)
2. In which quadrant is \(\tan\theta > 0\) and \(\cos\theta < 0\)?
A) QI
B) QII
C) QIII
D) QIV
3. Which positive angle is coterminal with \(-\frac{\pi}{4}\)?
A) \(\frac{3\pi}{4}\)
B) \(\frac{5\pi}{4}\)
C) \(\frac{7\pi}{4}\)
D) \(\frac{\pi}{4}\)
4. On the unit circle, which coordinate represents \(\sin\theta\)?
A) \(x\)
B) \(y\)
C) \(y/x\)
D) \(1/y\)
B Exact Trigonometric Values (2 pts each: 1 pt work/ref angle, 1 pt exact answer)
No decimals. Write exact simplified radicals.
5. \(\sin\left(\frac{4\pi}{3}\right)\) Radians
Quad: _____ | \(\theta'\): _____
Exact Value: ________
6. \(\cos(315^\circ)\) Degrees
Quad: _____ | \(\theta'\): _____
Exact Value: ________
7. \(\tan\left(\frac{5\pi}{6}\right)\) Radians
Quad: _____ | \(\theta'\): _____
Exact Value: ________
8. \(\sec(120^\circ)\) Reciprocal
Quad: _____ | \(\theta'\): _____
Exact Value: ________
9. \(\sin(\pi)\) Axis
Coordinates: \((x, y) = \) ________
Exact Value: ________
10. \(\cot\left(\frac{3\pi}{2}\right)\) Axis / Ratio
Ratio: \(\frac{\cos}{\sin} = \) ________
Exact Value: ________
C Free Response & Problem Solving (3 pts each)
Show all intermediate work clearly.
11. Evaluate: \(3\cos\left(\frac{\pi}{3}\right) - 2\sin\left(\frac{3\pi}{2}\right)\)
Substitute exact values for each term:
Final Result: ____________
12. Unit Circle Point \(P\left(x, -\frac{\sqrt{3}}{2}\right)\) in Quadrant IV
Find \(x > 0\) using \(x^2+y^2=1\), then determine \(\tan\theta\):
\(x = \) ________ | \(\tan\theta = \) ________
Bonus Challenge (+2 pts): Evaluate \(\sin^2(150^\circ) + \cos^2(330^\circ)\) exactly.
Bonus Answer: ____________
Unit Circle Mastery Quiz • Assessment Resource Page 1 of 1
Quiz Solutions Answer Key Quiz Solutions Answer Key
Form A Key
Unit Circle Mastery Quiz • Teacher Scoring Guide (25 Points Total)
Total: 25 Pts + 2 Bonus
A Part A Solutions: Concepts (1 pt each)
4 Points
1. Ref angle for \(150^\circ\)
QII: \(\theta' = 180^\circ - 150^\circ = 30^\circ\)
A) \(30^\circ\)
2. \(\tan > 0\) and \(\cos < 0\)
Tan (+) in QI, QIII; Cos (−) in QII, QIII
C) QIII
3. Coterminal with \(-\frac{\pi}{4}\)
\(-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}\)
C) \(\frac{7\pi}{4}\)
4. Coordinate for \(\sin\theta\)
On unit circle: \(x = \cos\theta, y = \sin\theta\)
B) \(y\)
B Part B Solutions: Exact Values (2 pts each: 1 pt ref/sign, 1 pt exact value)
12 Points
5. \(\sin\left(\frac{4\pi}{3}\right)\)
QIII \((-)\) | \(\theta' = \frac{\pi}{3}\)
Answer: \(-\frac{\sqrt{3}}{2}\)
6. \(\cos(315^\circ)\)
QIV \((+)\) | \(\theta' = 45^\circ\)
Answer: \(\frac{\sqrt{2}}{2}\)
7. \(\tan\left(\frac{5\pi}{6}\right)\)
QII \((-)\) | \(\theta' = \frac{\pi}{6}\)
Answer: \(-\frac{\sqrt{3}}{3}\)
8. \(\sec(120^\circ)\)
\(\cos(120^\circ) = -\frac{1}{2} \implies \frac{1}{\cos}\)
Answer: \(-2\)
9. \(\sin(\pi)\)
Negative x-axis \((-1, 0)\)
Answer: \(0\)
10. \(\cot\left(\frac{3\pi}{2}\right)\)
Point \((0, -1) \implies \frac{x}{y} = \frac{0}{-1}\)
Answer: \(0\)
C Part C Solutions: Free Response (3 pts each)
6 Points
11. Evaluate: \(3\cos\left(\frac{\pi}{3}\right) - 2\sin\left(\frac{3\pi}{2}\right)\)
\(\bullet\) \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) (1 pt)
\(\bullet\) \(\sin\left(\frac{3\pi}{2}\right) = -1\) (1 pt)
\(\bullet\) Calculation: \(3\left(\frac{1}{2}\right) - 2(-1) = \frac{3}{2} + 2 = \frac{7}{2}\) (1 pt)
Final Answer: \(\frac{7}{2}\) (or \(3.5\))
12. Unit Circle Point \(P\left(x, -\frac{\sqrt{3}}{2}\right)\) in QIV
\(\bullet\) \(x^2 + \left(-\frac{\sqrt{3}}{2}\right)^2 = 1 \implies x^2 = \frac{1}{4}\) (1 pt)
\(\bullet\) QIV \(\implies x > 0 \implies x = \frac{1}{2}\) (1 pt)
\(\bullet\) \(\tan\theta = \frac{-\sqrt{3}/2}{1/2} = -\sqrt{3}\) (1 pt)
Final Answer: \(x = \frac{1}{2}\) | \(\tan\theta = -\sqrt{3}\)
Bonus Solution: \(\sin^2(150^\circ) + \cos^2(330^\circ)\) Bonus Answer = \(1\)
\(\sin(150^\circ) = \frac{1}{2} \implies \left(\frac{1}{2}\right)^2 = \frac{1}{4}\). \(\cos(330^\circ) = \frac{\sqrt{3}}{2} \implies \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\). \(\frac{1}{4} + \frac{3}{4} = 1\). (Also satisfies \(\sin^2\theta + \cos^2\theta = 1\) since \(\cos(330^\circ) = \cos(30^\circ) = \cos(150^\circ)\) in magnitude).
Scale & Grade Equivalencies (25 pts):
A (90–100%): 23–25 pts
B (80–89%): 20–22 pts
C (70–79%): 17–19 pts
Reteach (<70%): \(\le\) 16 pts
Unit Circle Mastery Quiz Key • Teacher Resource Page 1 of 1
Unit Circle Study Flashcards Unit Circle Study Flashcards
Printable Cut-and-Fold Study Cards • Set 1: Values & Reciprocals
Cut solid outer border • Fold center dashed line
Card 1 • Front Degrees
\(\sin(150^\circ)\) & \(\cos(150^\circ)\)
What quadrant? Reference angle? Exact coordinates?
Fold backward to reveal answer
Card 1 • Solution QII \((- , +)\)
• Ref Angle: \(\theta' = 180^\circ - 150^\circ = 30^\circ\)
• \(\sin(150^\circ) = +\sin(30^\circ) = \frac{1}{2}\)
• \(\cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}\)
Coord: \((-\frac{\sqrt{3}}{2}, \frac{1}{2})\)
Card 2 • Front Degrees
\(\tan(225^\circ)\) & \(\cot(225^\circ)\)
What quadrant? Reference angle? Exact value?
Fold backward to reveal answer
Card 2 • Solution QIII \((- , -)\)
• Ref Angle: \(\theta' = 225^\circ - 180^\circ = 45^\circ\)
• In QIII, Tangent is Positive (+) via ASTC
• \(\tan(225^\circ) = +1\) | \(\cot(225^\circ) = \frac{1}{1} = 1\)
Coord: \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\)
Card 3 • Front Radians
\(\cos\left(\frac{5\pi}{3}\right)\) & \(\sin\left(\frac{5\pi}{3}\right)\)
What quadrant? Reference angle? Exact coordinates?
Fold backward to reveal answer
Card 3 • Solution QIV \((+ , -)\)
• Ref Angle: \(\theta' = 2\pi - \frac{5\pi}{3} = \frac{\pi}{3}\) (\(60^\circ\))
• \(\cos\left(\frac{5\pi}{3}\right) = +\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)
• \(\sin\left(\frac{5\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)
Coord: \((\frac{1}{2}, -\frac{\sqrt{3}}{2})\)
Card 4 • Front Radians
\(\sin\left(\frac{7\pi}{6}\right)\) & \(\tan\left(\frac{7\pi}{6}\right)\)
What quadrant? Reference angle? Rationalized exact values?
Fold backward to reveal answer
Card 4 • Solution QIII \((- , -)\)
• Ref Angle: \(\theta' = \frac{7\pi}{6} - \pi = \frac{\pi}{6}\) (\(30^\circ\))
• \(\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}\)
• \(\tan\left(\frac{7\pi}{6}\right) = +\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)
Coord: \((-\frac{\sqrt{3}}{2}, -\frac{1}{2})\)
Card 5 • Front Reciprocals
\(\sec(120^\circ)\) & \(\csc(210^\circ)\)
Evaluate base primary functions, then invert.
Fold backward to reveal answer
Card 5 • Solution Reciprocal Inversions
• \(\cos(120^\circ) = -\frac{1}{2} \implies \mathbf{\sec(120^\circ) = -2}\)
• \(\sin(210^\circ) = -\frac{1}{2} \implies \mathbf{\csc(210^\circ) = -2}\)
• Notice: Signs match primary functions exactly!
\(\sec = -2\) • \(\csc = -2\)
Unit Circle Study Flashcards • Math Manipulative Page 1 of 2
Unit Circle Study Flashcards
Printable Cut-and-Fold Study Cards • Set 2: Rules, Triangles & Shortcuts
Cards 6 – 10
Card 6 • Front Negative / Coterminal
\(\cos\left(-\frac{\pi}{4}\right)\) & \(\sin(420^\circ)\)
Find the positive coterminal angle \(\in [0, 2\pi)\) or \([0, 360^\circ)\).
Fold backward to reveal answer
Card 6 • Solution Coterminal Rules
• \(-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}\) (QIV) \(\implies \mathbf{\cos(-\frac{\pi}{4}) = +\frac{\sqrt{2}}{2}}\)
• \(420^\circ - 360^\circ = 60^\circ\) (QI) \(\implies \mathbf{\sin(420^\circ) = +\frac{\sqrt{3}}{2}}\)
• Remember: Cosine is even: \(\cos(-\theta) = \cos(\theta)\)!
Exact: \(\frac{\sqrt{2}}{2}\) • \(\frac{\sqrt{3}}{2}\)
Card 7 • Front Reference Angles
Reference Angle (\(\theta'\)) Rules
What are the formulas for \(\theta'\) in Quadrants I, II, III, and IV?
Fold backward to reveal answer
Card 7 • Solution Distance to x-axis
• QI: \(\theta' = \theta\)
• QII: \(180^\circ - \theta\) (\(\pi - \theta\))
• QIII: \(\theta - 180^\circ\) (\(\theta - \pi\))
• QIV: \(360^\circ - \theta\) (\(2\pi - \theta\))
Golden rule: Never measure to y-axis!
Card 8 • Front Sign Mnemonic
The ASTC Sign Rule
Which functions are positive in each quadrant?
Fold backward to reveal answer
Card 8 • Solution All Students Take Calculus
• Q I (A): ALL positive
• Q II (S): SINE positive
• Q III (T): TANGENT positive
• Q IV (C): COSINE positive
Reciprocals share exact same sign
Card 9 • Front Denominators
Radian Denominators Key
What angle families do denominators /6, /4, /3, /2 represent?
Fold backward to reveal answer
Card 9 • Solution Radian Rhythm
• /6 family: \(30^\circ\) family • coordinates: \((\pm\frac{\sqrt{3}}{2}, \pm\frac{1}{2})\)
• /4 family: \(45^\circ\) family • coordinates: \((\pm\frac{\sqrt{2}}{2}, \pm\frac{\sqrt{2}}{2})\)
• /3 family: \(60^\circ\) family • coordinates: \((\pm\frac{1}{2}, \pm\frac{\sqrt{3}}{2})\)
• /2 family: Quadrantal axes • coordinates: \((0, \pm 1)\)
Instant recognition shortcut
Card 10 • Front Memory Trick
The \(\frac{\sqrt{\text{Fingers}}}{2}\) Trick
How does the square root finger trick generate QI coordinates?
Fold backward to reveal answer
Card 10 • Solution Radical Sequence
• Cosine (counts down 4 to 0): \(\frac{\sqrt{4}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{0}}{2}\)
• Sine (counts up 0 to 4): \(\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\)
• Simplifies to: \(1, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0\)
For \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\)
Unit Circle Study Flashcards • Math Manipulative Page 2 of 2
Circle Check Exit Ticket Circle Check Exit Ticket
Formative Check • 5 Minutes
Name:
Date:
Period:
1. For angle \(\theta = \frac{7\pi}{4}\): Radians / Location
Quadrant:
Reference Angle (\(\theta'\)):
2. Evaluate exact values for \(\theta = \frac{7\pi}{4}\): ASTC • No Decimals
Evaluate \(\cos\left(\frac{7\pi}{4}\right)\):
Ans: ________
Evaluate \(\sin\left(\frac{7\pi}{4}\right)\):
Ans: ________
3. Error Analysis: A student claims \(\cos(150^\circ) = \frac{\sqrt{3}}{2}\). Spot the Error
Explain their mistake below and provide the correct exact value:
Rate your understanding today:
□ 1: Need Help □ 2: Getting There □ 3: Confident □ 4: Can Teach It
Cut along dashed line
Circle Check Exit Ticket
Formative Check • 5 Minutes
Name:
Date:
Period:
1. For angle \(\theta = \frac{7\pi}{4}\): Radians / Location
Quadrant:
Reference Angle (\(\theta'\)):
2. Evaluate exact values for \(\theta = \frac{7\pi}{4}\): ASTC • No Decimals
Evaluate \(\cos\left(\frac{7\pi}{4}\right)\):
Ans: ________
Evaluate \(\sin\left(\frac{7\pi}{4}\right)\):
Ans: ________
3. Error Analysis: A student claims \(\cos(150^\circ) = \frac{\sqrt{3}}{2}\). Spot the Error
Explain their mistake below and provide the correct exact value:
Rate your understanding today:
□ 1: Need Help □ 2: Getting There □ 3: Confident □ 4: Can Teach It
Reciprocal Values Guided Practice 1/x
Reciprocal Values Guided Practice
Evaluating Exact Values of \(\sec\theta\), \(\csc\theta\), and \(\cot\theta\) on the Unit Circle
Name:
Date:
Period:
1. Reciprocal Function Coordinate Definitions (On the unit circle with \(r=1\), where \(x = \cos\theta\) and \(y = \sin\theta\))
Secant Function \(\sec\theta\)
\(\sec\theta = \frac{1}{\cos\theta} = \frac{1}{x}\)
Partner: Cosine (\(x\)) Undef. when \(x = 0\)
Cosecant Function \(\csc\theta\)
\(\csc\theta = \frac{1}{\sin\theta} = \frac{1}{y}\)
Partner: Sine (\(y\)) Undef. when \(y = 0\)
Cotangent Function \(\cot\theta\)
\(\cot\theta = \frac{1}{\tan\theta} = \frac{x}{y}\)
Partner: Tangent (\(y/x\)) Undef. when \(y = 0\)
Sign Harmony (ASTC Rule)
A reciprocal fraction carries the exact same sign as its primary partner!
QI: All \(> 0\)
QII: \(\sin, \csc > 0\)
QIII: \(\tan, \cot > 0\)
QIV: \(\cos, \sec > 0\)
Radical Reciprocal Shortcuts (Rationalized) Flip & Rationalize
Primary
\(\frac{1}{2}\)
\(\rightarrow 2\)
Primary
\(\frac{\sqrt{2}}{2}\)
\(\rightarrow \sqrt{2}\)
Primary
\(\frac{\sqrt{3}}{2}\)
\(\rightarrow \frac{2\sqrt{3}}{3}\)
Primary
\(\frac{\sqrt{3}}{3}\)
\(\rightarrow \sqrt{3}\)
The 4-Step Exact Value Protocol for Reciprocals
Step 1: Partner & Quad Match \(\sec\rightarrow\cos\), \(\csc\rightarrow\sin\), \(\cot\rightarrow\tan\). Locate quadrant & note \(+/-\).
Step 2: Reference Angle Find acute \(\theta'\) to the x-axis: \(\theta' = \pi - \theta\), \(\theta - \pi\), or \(2\pi - \theta\).
Step 3: Primary Value Evaluate primary function at \(\theta'\) and attach sign: \(\text{Primary} = \pm f(\theta')\).
Step 4: Flip & Simplify Invert the fraction: \(\frac{1}{\text{value}}\). Rationalize radical denominators or write Undefined if 0.
Worked Exemplars (Teacher Models) Notice the progression from primary value to reciprocal
Model 1 \(\sec(120^\circ)\)
1. Partner & Quad: \(\cos\); \(120^\circ \in \text{QII} \rightarrow \cos < 0\)
2. Ref Angle: \(\theta' = 180^\circ - 120^\circ = 60^\circ\)
3. Primary: \(\cos(120^\circ) = -\cos(60^\circ) = -\frac{1}{2}\)
4. Invert: \(\sec(120^\circ) = \frac{1}{-1/2} = -2\)
\(\sec(120^\circ) = -2\)
Model 2 \(\csc\left(\frac{5\pi}{4}\right)\)
1. Partner & Quad: \(\sin\); \(\frac{5\pi}{4} \in \text{QIII} \rightarrow \sin < 0\)
2. Ref Angle: \(\theta' = \frac{5\pi}{4} - \pi = \frac{\pi}{4}\)
3. Primary: \(\sin\left(\frac{5\pi}{4}\right) = -\sin(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}\)
4. Invert: \(\frac{1}{-\sqrt{2}/2} = -\frac{2}{\sqrt{2}} = -\sqrt{2}\)
\(\csc\left(\frac{5\pi}{4}\right) = -\sqrt{2}\)
Model 3 \(\cot\left(\frac{11\pi}{6}\right)\)
1. Partner & Quad: \(\tan\); \(\frac{11\pi}{6} \in \text{QIV} \rightarrow \tan < 0\)
2. Ref Angle: \(\theta' = 2\pi - \frac{11\pi}{6} = \frac{\pi}{6}\)
3. Primary: \(\tan\left(\frac{11\pi}{6}\right) = -\frac{\sqrt{3}}{3} = -\frac{1}{\sqrt{3}}\)
4. Invert: \(\cot\left(\frac{11\pi}{6}\right) = -\sqrt{3}\)
\(\cot\left(\frac{11\pi}{6}\right) = -\sqrt{3}\)
Guided Practice: Complete the Scaffolded Steps Together
Fill in each guided blank carefully
Problem A: Evaluate \(\sec\left(\frac{7\pi}{6}\right)\) Radians / QIII
Step 1: Primary function is \(\cos\theta\). Quadrant is: Q______
Step 2: In this quadrant, cosine is \((+ \text{ or } -)\): Sign: ______
Step 3: Reference angle \(\theta' = \frac{7\pi}{6} - \pi =\): \(\theta' = \)______
Step 4: Primary value \(\cos\left(\frac{7\pi}{6}\right) =\): ____________
Step 5: Invert & rationalize: \(\sec\left(\frac{7\pi}{6}\right) = \frac{1}{\cos(7\pi/6)} =\): ____________
Final Rationalized Answer: \(\sec\left(\frac{7\pi}{6}\right) =\) ____________
Problem B: Evaluate \(\csc(300^\circ)\) Degrees / QIV
Step 1: Primary function is \(\sin\theta\). Quadrant is: Q______
Step 2: In QIV, sine is \((+ \text{ or } -)\): Sign: ______
Step 3: Reference angle \(\theta' = 360^\circ - 300^\circ =\): \(\theta' = \)______
Step 4: Primary value \(\sin(300^\circ) =\): ____________
Step 5: Invert & rationalize: \(\csc(300^\circ) = \frac{1}{\sin(300^\circ)} =\): ____________
Final Rationalized Answer: \(\csc(300^\circ) =\) ____________
Unit Circle Mastery • Reciprocal Trigonometric Functions Page 1 of 2
sec
Reciprocal Practice & Special Cases
Part 2: Multi-Quadrant Calculations, Quadrantal Extremes & Error Analysis
Independent Application
Part A: Evaluate Exact Values Using Reference Angles & Partners
Show all intermediate work in the designated columns
Problem
Quad & Sign
Ref Angle \(\theta'\)
Primary Partner Value
Exact Reciprocal Value
1. \(\sec(225^\circ)\)
Q: _____
Sign: ( )
| \(\theta' = \) _____ | \(\cos(225^\circ) =\) _______ | \(\sec(225^\circ) =\) ____________ |
| 2. \(\cot\left(\frac{2\pi}{3}\right)\) |
Q: _____
Sign: ( )
| \(\theta' = \) _____ | \(\tan\left(\frac{2\pi}{3}\right) =\) _______ | \(\cot\left(\frac{2\pi}{3}\right) =\) ____________ |
| 3. \(\csc(210^\circ)\) |
Q: _____
Sign: ( )
| \(\theta' = \) _____ | \(\sin(210^\circ) =\) _______ | \(\csc(210^\circ) =\) ____________ |
| 4. \(\sec\left(\frac{5\pi}{3}\right)\) |
Q: _____
Sign: ( )
| \(\theta' = \) _____ | \(\cos\left(\frac{5\pi}{3}\right) =\) _______ | \(\sec\left(\frac{5\pi}{3}\right) =\) ____________ |
| 5. \(\csc\left(\frac{3\pi}{4}\right)\) |
Q: _____
Sign: ( )
| \(\theta' = \) _____ | \(\sin\left(\frac{3\pi}{4}\right) =\) _______ | \(\csc\left(\frac{3\pi}{4}\right) =\) ____________ |
| 6. \(\cot(315^\circ)\) |
Q: _____
Sign: ( )
| \(\theta' = \) _____ | \(\tan(315^\circ) =\) _______ | \(\cot(315^\circ) =\) ____________ |
Part B: Quadrantal Angles & Undefined Values
Recall: Division by zero (\(\frac{1}{0}\)) is Undefined
7. \(\sec(180^\circ)\)
Coord: \((x,y) = (-1, 0)\)
\(\cos(180^\circ) =\) ________
Answer: ________
8. \(\csc(\pi)\)
Coord: \((x,y) = (-1, 0)\)
\(\sin(\pi) =\) ________
Answer: ________
9. \(\cot\left(\frac{\pi}{2}\right)\)
Coord: \((x,y) = (0, 1)\)
\(\frac{x}{y} = \frac{0}{1} =\) ________
Answer: ________
10. \(\sec\left(\frac{3\pi}{2}\right)\)
Coord: \((x,y) = (0, -1)\)
\(\cos(3\pi/2) =\) ________
Answer: ________
Part C: Negative & Coterminal Rotations
Find positive coterminal angle \(\theta \in [0, 2\pi)\) or \([0^\circ, 360^\circ)\) first
11. \(\sec\left(-\frac{\pi}{3}\right)\) Clockwise rotation
Coterminal: \(2\pi - \frac{\pi}{3} = \) ________
Quadrant: ________ | Sign: ( )
Answer: ____________
12. \(\csc(-240^\circ)\) Negative degrees
Coterminal: \(-240^\circ + 360^\circ = \) ________
Quadrant: ________ | Sign: ( )
Answer: ____________
13. \(\cot\left(\frac{13\pi}{6}\right)\) \(\theta > 2\pi\)
Coterminal: \(\frac{13\pi}{6} - 2\pi = \) ________
Quadrant: ________ | Sign: ( )
Answer: ____________
Part D: Error Analysis — "The Mistake Buster"
Identify & Correct
A student's solution for \(\sec(150^\circ)\):
"Step 1: Partner of \(\sec\) is \(\sin\), so find \(\sin(150^\circ)\)."
"Step 2: \(\theta' = 30^\circ\), and in QII, \(\sin(150^\circ) = +\frac{1}{2}\)."
"Step 3: Inverting gives \(\sec(150^\circ) = 2\)."
Describe the student's error and provide the correct exact value:
Error: ________________________________________________
Correct Value of \(\sec(150^\circ)\): ______________
Unit Circle Mastery • Reciprocal Trigonometric Functions Page 2 of 2
Reciprocal Values Answer Key KEY
Reciprocal Values Answer Key
Teacher Edition • Complete Worked Solutions & Pedagogical Notes
Teacher Reference
Instructional Coaching & Common Reciprocal Pitfalls
1. Partner Pairing Trap Students instinctively pair "s with s" (\(\sec\leftrightarrow\sin\)). Reinforce that the letters alternate: \(\sec\leftrightarrow\cos\) and \(\csc\leftrightarrow\sin\).
2. Sign Harmony Rule Remind students that flipping a fraction never changes its sign! If \(\cos\theta < 0\) in QII, then \(\sec\theta < 0\) in QII.
3. Rationalizing Fluency Teach \(\frac{2}{\sqrt{3}} \rightarrow \frac{2\sqrt{3}}{3}\) and \(\frac{2}{\sqrt{2}} \rightarrow \sqrt{2}\) as instant reflexes to save student calculation time.
Page 1 "We Do" Guided Practice Solutions
Step-by-Step Breakdown
Problem A: \(\sec\left(\frac{7\pi}{6}\right)\) QIII Radians
Step 1: Primary function & Quadrant: Partner: \(\cos\); Quadrant III
Step 2: Sign in Quadrant III: Negative \((-)\)
Step 3: Reference angle (\(\theta'\)): \(\theta' = \frac{7\pi}{6} - \pi = \frac{\pi}{6}\)
Step 4: Primary partner value: \(\cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\)
Step 5: Reciprocal & rationalize: \(\frac{1}{-\sqrt{3}/2} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\)
Final Answer: \(\sec\left(\frac{7\pi}{6}\right) = -\frac{2\sqrt{3}}{3}\)
Problem B: \(\csc(300^\circ)\) QIV Degrees
Step 1: Primary function & Quadrant: Partner: \(\sin\); Quadrant IV
Step 2: Sign in Quadrant IV: Negative \((-)\)
Step 3: Reference angle (\(\theta'\)): \(\theta' = 360^\circ - 300^\circ = 60^\circ\)
Step 4: Primary partner value: \(\sin(300^\circ) = -\frac{\sqrt{3}}{2}\)
Step 5: Reciprocal & rationalize: \(\frac{1}{-\sqrt{3}/2} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\)
Final Answer: \(\csc(300^\circ) = -\frac{2\sqrt{3}}{3}\)
First Quadrant Reciprocal Reference Standards (\(0^\circ\) to \(90^\circ\))
\(0^\circ\) (\(0\))
\(\sec(0) = 1\)
\(\csc(0) = \text{Undef}\)
\(\cot(0) = \text{Undef}\)
\(30^\circ\) (\(\pi/6\))
\(\sec = \frac{2\sqrt{3}}{3}\)
\(\csc = 2\)
\(\cot = \sqrt{3}\)
\(45^\circ\) (\(\pi/4\))
\(\sec = \sqrt{2}\)
\(\csc = \sqrt{2}\)
\(\cot = 1\)
\(60^\circ\) (\(\pi/3\))
\(\sec = 2\)
\(\csc = \frac{2\sqrt{3}}{3}\)
\(\cot = \frac{\sqrt{3}}{3}\)
\(90^\circ\) (\(\pi/2\))
\(\sec = \text{Undef}\)
\(\csc = 1\)
\(\cot = 0\)
Unit Circle Mastery • Reciprocal Practice Solutions Page 1 of 2
P2
Independent Practice Solutions (Parts A – D)
Verified Solutions, Intermediate Steps & Error Correction Rubric
Complete Key
Part A Solutions: Reference Angle & Value Table
Problem
Quad & Sign
Ref Angle \(\theta'\)
Primary Partner Value
Exact Reciprocal Value
1. \(\sec(225^\circ)\)
QIII, \((-)\)
\(45^\circ\)
\(\cos(225^\circ) = -\frac{\sqrt{2}}{2}\)
\(-\sqrt{2}\)
2. \(\cot\left(\frac{2\pi}{3}\right)\)
QII, \((-)\)
\(\frac{\pi}{3}\)
\(\tan\left(\frac{2\pi}{3}\right) = -\sqrt{3}\)
\(-\frac{\sqrt{3}}{3}\)
3. \(\csc(210^\circ)\)
QIII, \((-)\)
\(30^\circ\)
\(\sin(210^\circ) = -\frac{1}{2}\)
\(-2\)
4. \(\sec\left(\frac{5\pi}{3}\right)\)
QIV, \((+)\)
\(\frac{\pi}{3}\)
\(\cos\left(\frac{5\pi}{3}\right) = +\frac{1}{2}\)
\(+2\)
5. \(\csc\left(\frac{3\pi}{4}\right)\)
QII, \((+)\)
\(\frac{\pi}{4}\)
\(\sin\left(\frac{3\pi}{4}\right) = +\frac{\sqrt{2}}{2}\)
\(+\sqrt{2}\)
6. \(\cot(315^\circ)\)
QIV, \((-)\)
\(45^\circ\)
\(\tan(315^\circ) = -1\)
\(-1\)
Part B Solutions: Quadrantal Angles & Division by Zero
7. \(\sec(180^\circ)\)
\((x,y) = (-1, 0)\)
\(-1\)
8. \(\csc(\pi)\)
\((x,y) = (-1, 0)\)
Undefined
9. \(\cot\left(\frac{\pi}{2}\right)\)
\((x,y) = (0, 1)\)
\(0\) (Not Undef!)
10. \(\sec\left(\frac{3\pi}{2}\right)\)
\((x,y) = (0, -1)\)
Undefined
Part C Solutions: Negative & Multi-Rotation Angles
11. \(\sec\left(-\frac{\pi}{3}\right)\)
Coterminal: \(2\pi - \frac{\pi}{3} = \frac{5\pi}{3}\) (QIV)
Cosine is positive in QIV.
\(\cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)
\(\sec\left(-\frac{\pi}{3}\right) = 2\)
12. \(\csc(-240^\circ)\)
Coterminal: \(-240^\circ + 360^\circ = 120^\circ\) (QII)
Sine is positive in QII (\(\theta'=60^\circ\)).
\(\sin(120^\circ) = +\frac{\sqrt{3}}{2}\)
\(\csc(-240^\circ) = \frac{2\sqrt{3}}{3}\)
13. \(\cot\left(\frac{13\pi}{6}\right)\)
Coterminal: \(\frac{13\pi}{6} - 2\pi = \frac{\pi}{6}\) (QI)
All functions are positive in QI.
\(\tan\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3} = \frac{1}{\sqrt{3}}\)
\(\cot\left(\frac{13\pi}{6}\right) = \sqrt{3}\)
Part D Solution & Grading Rubric: "The Mistake Buster"
Identification of Student Misconceptions:
Error 1 (Wrong Partner): The student matched \(\sec\) with \(\sin\). Secant is defined as the reciprocal of cosine : \(\sec\theta = \frac{1}{\cos\theta}\).Error 2 (Sign Error): In Quadrant II (\(150^\circ\)), cosine is negative (\(-\)), meaning secant must also be negative, not positive!
Correct Worked Solution:
1. \(\cos(150^\circ) \in \text{QII} \rightarrow \cos < 0\)
2. \(\theta' = 180^\circ - 150^\circ = 30^\circ\)
3. \(\cos(150^\circ) = -\frac{\sqrt{3}}{2}\)
4. \(\sec(150^\circ) = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\)
Exact Value: \(-\frac{2\sqrt{3}}{3}\)
Unit Circle Mastery • Reciprocal Practice Solutions Page 2 of 2
Circle Values Station Activity Circle Values Station Activity
Unit Circle Mastery • Collaborative Trigonometry Investigation & Speed Stations
Cooperative Stations
Station Rotation Protocol (8–10 Minutes Per Station) Rotate clockwise at the chime • Record all work on Page 4 Passport
1. Facilitator: Reads prompt aloud and keeps time.
2. Resource Lead: Checks unit circle definitions & ASTC rules.
3. Verifier: Validates signs, fractions, and reciprocals.
4. Reporter: Shares team consensus on the final passport.
1
Station 1: Rotation & Coterminal Compass
Standard Angles \([0, 360^\circ)\) & \([0, 2\pi)\)
Angles in standard position wrap around the origin infinitely. Two angles are coterminal if they share the same terminal ray: add or subtract multiples of \(360^\circ\) or \(2\pi\).
Task 1A: Degree Coterminals
Find a coterminal angle \(\alpha \in [0, 360^\circ)\) and identify its quadrant:
• \(\theta_1 = -240^\circ \rightarrow \alpha = \) ? (Quadrant: ?)
• \(\theta_2 = 570^\circ \rightarrow \alpha = \) ? (Quadrant: ?)
Goal: Add or subtract multiples of \(360^\circ\).
Task 1B: Radian Coterminals
Find a coterminal angle \(\alpha \in [0, 2\pi)\) and identify its quadrant:
• \(\theta_3 = \frac{19\pi}{6} \rightarrow \alpha = \) ? (Quadrant: ?)
• \(\theta_4 = -\frac{9\pi}{4} \rightarrow \alpha = \) ? (Quadrant: ?)
Goal: Subtract/add multiples of \(2\pi = \frac{12\pi}{6}\) or \(\frac{8\pi}{4}\).
2
Station 2: Reference Angle & ASTC Radar
Acute Reference \(\theta'\) & All Students Take Calculus
The reference angle \(\theta'\) is always the positive acute angle measured to the x-axis. Combine \(\theta'\) with the quadrant sign rule (ASTC) to evaluate any angle.
Task 2A: Acute \(\theta'\) Search
Calculate the exact reference angle \(\theta'\) for each given angle:
• \(\theta_1 = 225^\circ \rightarrow \theta' = \) ?
• \(\theta_2 = \frac{7\pi}{4} \rightarrow \theta' = \) ?
• \(\theta_3 = \frac{5\pi}{3} \rightarrow \theta' = \) ?
Rule: \(\theta'\) is always acute (\(0 < \theta' \le 90^\circ\) or \(\frac{\pi}{2}\)).
Task 2B: Quadrant Sign Matrix
For the angle \(\theta = \frac{5\pi}{6}\), specify the quadrant and determine whether each is Positive (+) or Negative (-):
\(\sin\theta\): ___
\(\cos\theta\): ___
\(\tan\theta\): ___
\(\sec\theta\): ___
ASTC: Quadrant II has only Sine & Cosecant positive.
Circle Values Station Activity • Stations 1 & 2 Task Guide Page 1 of 4
Exact Value Sprint & Reciprocal Realm
Unit Circle Mastery • Station 3 & Station 4 Challenges
Stations 3 & 4
3
Station 3: Exact Value Speed Sprint (Primary Functions)
No Decimals • Simplified Radicals
On the unit circle, \(\cos\theta = x\), \(\sin\theta = y\), and \(\tan\theta = \frac{y}{x}\). Use reference angles and quadrant signs to find the exact values rapidly without a calculator.
A Task 3A: Degree Special Values
State quadrant, reference angle, and exact simplified value:
1. \(\sin(210^\circ)\) Quad III, \(\theta'=30^\circ\)
2. \(\cos(135^\circ)\) Quad II, \(\theta'=45^\circ\)
3. \(\tan(240^\circ)\) Quad III, \(\theta'=60^\circ\)
B Task 3B: Radian Special Values
Evaluate each exact primary trig value:
4. \(\sin\left(\frac{5\pi}{6}\right)\) Quad II, \(\theta'=\frac{\pi}{6}\)
5. \(\cos\left(\frac{7\pi}{4}\right)\) Quad IV, \(\theta'=\frac{\pi}{4}\)
6. \(\tan\left(\frac{4\pi}{3}\right)\) Quad III, \(\theta'=\frac{\pi}{3}\)
4
Station 4: Reciprocal Function Gauntlet (\(\sec, \csc, \cot\))
Reciprocal & Undefined Rules
Reciprocal identities: \(\sec\theta = \frac{1}{\cos\theta} = \frac{1}{x}\), \(\csc\theta = \frac{1}{\sin\theta} = \frac{1}{y}\), and \(\cot\theta = \frac{1}{\tan\theta} = \frac{x}{y}\). Never leave a 0 in the denominator (value is undefined); radical answers do not require rationalizing.
Task 4A: Degree Reciprocals
Evaluate exact values:
• \(\csc(30^\circ) = \frac{1}{\sin(30^\circ)}\)
• \(\sec\left(120^\circ\right) = \frac{1}{\cos(120^\circ)}\)
• \(\cot(225^\circ) = \frac{1}{\tan(225^\circ)}\)
Flip the primary value; keep the same sign!
Task 4B: Radian Reciprocals
Find the exact reciprocal value (no rationalizing needed):
• \(\sec\left(\frac{11\pi}{6}\right) = \frac{1}{\cos(11\pi/6)}\)
• \(\csc\left(-\frac{\pi}{3}\right) = \frac{1}{\sin(-\pi/3)}\)
• \(\cot\left(\frac{\pi}{2}\right) = \frac{\cos(\pi/2)}{\sin(\pi/2)}\)
Flip the primary value: \(\sec\theta = \frac{1}{\cos\theta}\).
Circle Values Station Activity • Stations 3 & 4 Task Guide Page 2 of 4
Expressions & Coordinate Mysteries
Unit Circle Mastery • Station 5 & Station 6 Challenges
Stations 5 & 6
5
Station 5: Expression Arena
Exact Linear Evaluations
Substitute the exact numerical value of each trigonometric expression first, apply operations carefully, and simplify algebraically. Do not approximate with decimals.
Task 5A: Linear Evaluation (Set 1)
Evaluate the exact numerical value:
\(2\cos(60^\circ) + 4\sin(30^\circ) - \tan(135^\circ)\)
Substitute \(\cos(60^\circ) = \frac{1}{2}\), \(\sin(30^\circ) = \frac{1}{2}\), and \(\tan(135^\circ) = -1\).
Team Check: Pay special attention to subtracting a negative.
Task 5B: Linear Evaluation (Set 2)
Evaluate the exact numerical value:
\(4\cos(60^\circ) - 2\sin(210^\circ) + \tan(225^\circ)\)
Substitute \(\cos(60^\circ) = \frac{1}{2}\), \(\sin(210^\circ) = -\frac{1}{2}\), and \(\tan(225^\circ) = 1\).
Team Check: Carefully multiply coefficients before combining.
6
Station 6: Coordinate Mystery Detective
Unit Circle Equation \(x^2 + y^2 = 1\)
Every point on the unit circle satisfies \(x^2 + y^2 = 1\). Use algebraic equations and given quadrant constraints to solve for missing coordinates and determine exact trigonometric ratios.
Clue 6A: Coordinate Search in Quad II
Point \(P\left(-\frac{1}{2}, y\right)\) is located on the unit circle in Quadrant II.
Solve for the exact value of \(y > 0\) using \(x^2 + y^2 = 1\).
Write the exact values of \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\).
\(\left(-\frac{1}{2}\right)^2 + y^2 = 1 \implies \frac{1}{4} + y^2 = 1\)
Clue 6B: Coordinate Search in Quad III
Point \(Q\left(x, -\frac{1}{2}\right)\) is located on the unit circle in Quadrant III.
Solve for the exact value of \(x < 0\) using \(x^2 + y^2 = 1\).
Write the exact values of \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\).
\(x^2 + \left(-\frac{1}{2}\right)^2 = 1 \implies x^2 + \frac{1}{4} = 1\)
Circle Values Station Activity • Stations 5 & 6 Task Guide Page 3 of 4
Student Station Passport
Unit Circle Mastery • Official Team Work & Solution Log
Name:
Date:
Station # Start:
1 Station 1: Coterminal Compass Passport Stamp: [ ]
1A: \(-240^\circ \rightarrow\) Coterm: ________ (Quad: ____) | \(570^\circ \rightarrow\) Coterm: ________ (Quad: ____)
1B: \(\frac{19\pi}{6} \rightarrow\) Coterm: ________ (Quad: ____) | \(-\frac{9\pi}{4} \rightarrow\) Coterm: ________ (Quad: ____)
2 Station 2: Reference & ASTC Radar Passport Stamp: [ ]
2A: \(225^\circ \rightarrow \theta'=\) _______ | \(\frac{7\pi}{4} \rightarrow \theta'=\) _______ | \(\frac{5\pi}{3} \rightarrow \theta'=\) _______
2B: \(\theta = \frac{5\pi}{6}\) (Quad: ____) \(\sin:\) ___ \(\cos:\) ___ \(\tan:\) ___ \(\sec:\) ___
3 Station 3: Exact Value Sprint Passport Stamp: [ ]
\(\sin(210^\circ)=\) _____
\(\cos(135^\circ)=\) _____
\(\tan(240^\circ)=\) _____
\(\sin\left(\frac{5\pi}{6}\right)=\) _____
\(\cos\left(\frac{7\pi}{4}\right)=\) _____
\(\tan\left(\frac{4\pi}{3}\right)=\) _____
4 Station 4: Reciprocal Gauntlet Passport Stamp: [ ]
\(\csc(30^\circ)=\) _____
\(\sec(120^\circ)=\) _____
\(\cot(225^\circ)=\) _____
\(\sec\left(\frac{11\pi}{6}\right)=\) _____
\(\csc\left(-\frac{\pi}{3}\right)=\) _____
\(\cot\left(\frac{\pi}{2}\right)=\) _____
5 Station 5: Expression Arena Passport Stamp: [ ]
5A: \(2\cos(60^\circ) + 4\sin(30^\circ) - \tan(135^\circ) = \) ___________
5B: \(4\cos(60^\circ) - 2\sin(210^\circ) + \tan(225^\circ) = \) ___________
6 Station 6: Coordinate Mystery Detective Passport Stamp: [ ]
6A: \(y = \) ________ | \(\sin\theta = \) ________ | \(\tan\theta = \) ________
6B: \(x = \) ________ | \(\cos\theta = \) ________ | \(\tan\theta = \) ________
Team Reflection & Strategy Debrief Complete before turning in
1. Which station was the most mathematically challenging for your team, and what specific strategy or rule helped you solve it?
Team Members Signatures: ____________________________ , ____________________________ Final Score: _____ / 30
Circle Values Station Activity • Student Recording Passport Page 4 of 4
Station Activity Answer Key Station Activity Answer Key
Unit Circle Mastery • Teacher Solutions & Station Facilitation Guide
Complete Solutions • 30 Pts
Facilitation & Spot-Checking Strategy Pacing: 8–10 min rotations • 5 pts per station (2.5 pts per task)
Circulate with a stamp or highlighter. Each station features exactly two focused tasks (Task A and Task B). Emphasize quadrant sign reasoning and exact radical reciprocals without decimal approximations.
1
Station 1 Solutions: Rotation & Coterminal Compass (5 Pts)
Standard Angles
Task 1A: Degree Coterminals (2.5 Pts)
• \(-240^\circ + 360^\circ =\) \(120^\circ\) (Quadrant II)
• \(570^\circ - 360^\circ =\) \(210^\circ\) (Quadrant III)
Score: 1.25 pts per coterminal angle + quadrant identification.
Task 1B: Radian Coterminals (2.5 Pts)
• \(\frac{19\pi}{6} - \frac{12\pi}{6} =\) \(\frac{7\pi}{6}\) (Quadrant III)
• \(-\frac{9\pi}{4} + \frac{16\pi}{4} =\) \(\frac{7\pi}{4}\) (Quadrant IV)
Added two rotations \(2(2\pi) = \frac{16\pi}{4}\) for negative angle.
2
Station 2 Solutions: Reference Angle & ASTC Radar (5 Pts)
\(\theta'\) & Signs
Task 2A: Reference Angles (2.5 Pts)
• \(225^\circ \rightarrow \theta' = 225^\circ - 180^\circ =\) \(45^\circ\)
• \(\frac{7\pi}{4} \rightarrow \theta' = 2\pi - \frac{7\pi}{4} =\) \(\frac{\pi}{4}\)
• \(\frac{5\pi}{3} \rightarrow \theta' = 2\pi - \frac{5\pi}{3} =\) \(\frac{\pi}{3}\)
Reference angles must always be acute and positive to the x-axis.
Task 2B: Quadrant II Signs (2.5 Pts)
For \(\theta = \frac{5\pi}{6}\) in Quadrant II:
\(\sin\theta\): (+) Positive
\(\cos\theta\): (-) Negative
\(\tan\theta\): (-) Negative
\(\sec\theta\): (-) Negative
ASTC rule: Only Sine and Cosecant are positive in QII.
3
Station 3 Solutions: Exact Value Speed Sprint (5 Pts)
Primary Values
Task 3A: Degree Special Values (2.5 Pts)
1. \(\sin(210^\circ) =\) \(-\frac{1}{2}\) (QIII)
2. \(\cos(135^\circ) =\) \(-\frac{\sqrt{2}}{2}\) (QII)
3. \(\tan(240^\circ) =\) \(\sqrt{3}\) (QIII)
ASTC signs applied to reference angles 30°, 45°, 60°.
Task 3B: Radian Special Values (2.5 Pts)
4. \(\sin\left(\frac{5\pi}{6}\right) =\) \(\frac{1}{2}\) (QII)
5. \(\cos\left(\frac{7\pi}{4}\right) =\) \(\frac{\sqrt{2}}{2}\) (QIV)
6. \(\tan\left(\frac{4\pi}{3}\right) =\) \(\sqrt{3}\) (QIII)
Positive in Q2, Q4, Q3 respectively.
Circle Values Station Activity • Teacher Answer Key Page 1 of 2
II
Stations 4–6 Solutions & Team Assessment Rubric
Precalculus Mastery Standard
4
Station 4 Solutions: Reciprocal Function Gauntlet (5 Pts)
Reciprocals
Task 4A: Degree Reciprocals (2.5 Pts)
• \(\csc(30^\circ) = \frac{1}{\sin(30^\circ)} = \frac{1}{1/2} =\) \(2\)
• \(\sec(120^\circ) = \frac{1}{\cos(120^\circ)} = \frac{1}{-1/2} =\) \(-2\)
• \(\cot(225^\circ) = \frac{1}{\tan(225^\circ)} = \frac{1}{1} =\) \(1\)
Reciprocals inherit the exact sign of their base function.
Task 4B: Radian Reciprocals (2.5 Pts)
• \(\sec\left(\frac{11\pi}{6}\right) = \frac{1}{\sqrt{3}/2} =\) \(\frac{2}{\sqrt{3}}\) (\(\frac{2\sqrt{3}}{3}\) also ok)
• \(\csc\left(-\frac{\pi}{3}\right) = \frac{1}{-\sqrt{3}/2} =\) \(-\frac{2}{\sqrt{3}}\) (\(-\frac{2\sqrt{3}}{3}\) also ok)
• \(\cot\left(\frac{\pi}{2}\right) = \frac{\cos(\pi/2)}{\sin(\pi/2)} = \frac{0}{1} =\) \(0\)
Unrationalized radical expressions are fully credited.
5
Station 5 Solutions: Expression Arena (5 Pts)
Linear Evaluations
Task 5A: Linear Evaluation (Set 1 - 2.5 Pts)
\(= 2\cos(60^\circ) + 4\sin(30^\circ) - \tan(135^\circ)\)
Subtracting \(\tan(135^\circ) = -1\) yields \(+1\).
Task 5B: Linear Evaluation (Set 2 - 2.5 Pts)
\(= 4\cos(60^\circ) - 2\sin(210^\circ) + \tan(225^\circ)\)
\(-2 \cdot (-1/2) = +1\). Both tasks evaluate cleanly to \(4\).
6
Station 6 Solutions: Coordinate Mystery Detective (5 Pts)
Reconstruction
Clue 6A: Coordinate Search in Quad II (2.5 Pts)
• Equation: \(\left(-\frac{1}{2}\right)^2 + y^2 = 1 \implies \frac{1}{4} + y^2 = 1 \implies y = \) \(\frac{\sqrt{3}}{2}\)
• \(\sin\theta = y = \) \(\frac{\sqrt{3}}{2}\)
• \(\cos\theta = x = \) \(-\frac{1}{2}\)
• \(\tan\theta = \frac{y}{x} = \frac{\sqrt{3}/2}{-1/2} = \) \(-\sqrt{3}\)
Clue 6B: Coordinate Search in Quad III (2.5 Pts)
• Equation: \(x^2 + \left(-\frac{1}{2}\right)^2 = 1 \implies x^2 + \frac{1}{4} = 1 \implies x = \) \(-\frac{\sqrt{3}}{2}\)
• \(\sin\theta = y = \) \(-\frac{1}{2}\)
• \(\cos\theta = x = \) \(-\frac{\sqrt{3}}{2}\)
• \(\tan\theta = \frac{y}{x} = \frac{-1/2}{-\sqrt{3}/2} = \) \(\frac{1}{\sqrt{3}}\) (\(\frac{\sqrt{3}}{3}\))
Mastery Scoring Rubric (30 Total Points) 5 Points Per Station (2.5 Pts Per Task)
Score Tier
Mathematical Accuracy
Team Collaboration & Passport Evidence
27–30 pts (Advanced)
All 6 stations fully correct; exact radical values identified, signs accurate, no decimal approximations.
Clear work shown in every passport box; reflective prompt completed thoughtfully.
23–26 pts (Proficient)
Minor sign error in 1–2 items; core unit circle values and reciprocal definitions solid.
Work shown in all stations; minor gaps in intermediate algebraic substitution.
18–22 pts (Developing)
Consistent sign errors across quadrants (ASTC misapplied) or confusion with coordinate signs.
Several blank sub-problems; incomplete team debrief section.
Circle Values Station Activity • Teacher Answer Key Page 2 of 2
Circle Stations Table Tents 1
Unit Circle Mastery • Station 1
Rotation & Coterminal Compass
Table Tent
Anchor Formulas: Coterminal Rotations
Degree Rule: \(\theta_{\text{coterm}} = \theta \pm 360^\circ \cdot k\)
Radian Rule: \(\theta_{\text{coterm}} = \theta \pm 2\pi \cdot k\)
Standard Interval Boundaries:
QI: \([0, 90^\circ)\) or \([0, \frac{\pi}{2})\)
QII: \([90^\circ, 180^\circ)\) or \([\frac{\pi}{2}, \pi)\)
QIII: \([180^\circ, 270^\circ)\) or \([\pi, \frac{3\pi}{2})\)
QIV: \([270^\circ, 360^\circ)\) or \([\frac{3\pi}{2}, 2\pi)\)
Station Protocol: Add or subtract full rotations until angle lies in standard range! 8–10 Min Limit
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Station 1 Problems: Solve on Your Team Passport
Page 4 Log
Task 1A: Degree Coterminals
Find a coterminal angle \(\alpha \in [0, 360^\circ)\) and identify its quadrant:
1. \(\theta_1 = -240^\circ\)
2. \(\theta_2 = 570^\circ\)
Record angle and quadrant in Passport box 1A.
Task 1B: Radian Coterminals
Find a coterminal angle \(\alpha \in [0, 2\pi)\) and identify its quadrant:
3. \(\theta_3 = \frac{19\pi}{6}\)
4. \(\theta_4 = -\frac{9\pi}{4}\)
Use common denominators: \(2\pi = \frac{12\pi}{6} = \frac{8\pi}{4}\).
Unit Circle Table Tent 1 • Fold in half along the dashed line Check work with teammate before rotating
2
Unit Circle Mastery • Station 2
Reference Angle & ASTC Radar
Table Tent
Reference Angle Rules (\(\theta'\) is acute, positive, to the x-axis)
Quad I \(\theta' = \theta\)
Quad II \(\theta' = 180^\circ - \theta\)
Quad III \(\theta' = \theta - 180^\circ\)
Quad IV \(\theta' = 360^\circ - \theta\)
ASTC Sign Harmonizer:
A: All Positive (QI)
S: Sine Positive (QII)
T: Tan Positive (QIII)
C: Cos Positive (QIV)
Station Protocol: Measure to the horizontal x-axis, NEVER the y-axis! 8–10 Min Limit
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Station 2 Problems: Solve on Your Team Passport
Page 4 Log
Task 2A: Acute \(\theta'\) Search
Calculate the exact acute reference angle \(\theta'\) for each:
1. \(\theta_1 = 225^\circ \rightarrow \theta' =\) ?
2. \(\theta_2 = \frac{7\pi}{4} \rightarrow \theta' =\) ?
3. \(\theta_3 = \frac{5\pi}{3} \rightarrow \theta' =\) ?
Remember: \(0 < \theta' \le 90^\circ\) or \(\frac{\pi}{2}\).
Task 2B: Quadrant Sign Matrix
For \(\theta = \frac{5\pi}{6}\), state the quadrant and determine if each is (+) or (-):
\(\sin\theta:\) _____
\(\cos\theta:\) _____
\(\tan\theta:\) _____
\(\sec\theta:\) _____
Secant has the same sign as Cosine.
Unit Circle Table Tent 2 • Fold in half along the dashed line Check work with teammate before rotating
3
Unit Circle Mastery • Station 3
Exact Value Speed Sprint
Table Tent
Quadrant I Base Coordinates Reference Map
\(30^\circ\) or \(\frac{\pi}{6}\) \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
\(45^\circ\) or \(\frac{\pi}{4}\) \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)
\(60^\circ\) or \(\frac{\pi}{3}\) \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
Fundamental Relationship on the Circle:
\(\cos\theta = x\)
\(\sin\theta = y\)
\(\tan\theta = \frac{y}{x}\)
Station Protocol: No calculators allowed! Simplify all fractions and radicals completely. 8–10 Min Limit
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Station 3 Problems: Solve on Your Team Passport
Page 4 Log
Task 3A: Degree Special Values
Evaluate each exact primary trig value:
1. \(\sin(210^\circ) =\) _____
2. \(\cos(135^\circ) =\) _____
3. \(\tan(240^\circ) =\) _____
Identify reference angle and apply ASTC quadrant signs.
Task 3B: Radian Special Values
Evaluate each exact primary trig value:
4. \(\sin\left(\frac{5\pi}{6}\right) =\) _____
5. \(\cos\left(\frac{7\pi}{4}\right) =\) _____
6. \(\tan\left(\frac{4\pi}{3}\right) =\) _____
Keep radicals exact: do not write decimal approximations.
Unit Circle Table Tent 3 • Fold in half along the dashed line Check work with teammate before rotating
4
Unit Circle Mastery • Station 4
Reciprocal Function Gauntlet
Table Tent
Reciprocal Identities & Rules
Cosecant \(\csc\theta = \frac{1}{\sin\theta} = \frac{1}{y}\)
Secant \(\sec\theta = \frac{1}{\cos\theta} = \frac{1}{x}\)
Cotangent \(\cot\theta = \frac{1}{\tan\theta} = \frac{x}{y}\)
Golden Rules for Reciprocals:
Sign Harmony: Reciprocals ALWAYS have the exact same sign (+ or -) as their primary functions.
Division by Zero: \(\frac{1}{0}\) is UNDEFINED. However, \(\frac{0}{1} = 0\).
Radical Form: Exact unrationalized radical answers (like \(\frac{2}{\sqrt{3}}\)) are accepted!
Station Protocol: To evaluate, find the primary trig value first, then invert the fraction! 8–10 Min Limit
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Station 4 Problems: Solve on Your Team Passport
Page 4 Log
Task 4A: Degree Reciprocals
Evaluate each exact reciprocal value:
1. \(\csc(30^\circ) = \frac{1}{\sin(30^\circ)}\)
2. \(\sec(120^\circ) = \frac{1}{\cos(120^\circ)}\)
3. \(\cot(225^\circ) = \frac{1}{\tan(225^\circ)}\)
Flip the primary fraction; keep the sign.
Task 4B: Radian Reciprocals
Find each exact value (unrationalized form accepted):
4. \(\sec\left(\frac{11\pi}{6}\right) = \frac{1}{\cos(11\pi/6)}\)
5. \(\csc\left(-\frac{\pi}{3}\right) = \frac{1}{\sin(-\pi/3)}\)
6. \(\cot\left(\frac{\pi}{2}\right) = \frac{\cos(\pi/2)}{\sin(\pi/2)}\)
Notice that \(\cot(\pi/2) = 0/1 = 0\).
Unit Circle Table Tent 4 • Fold in half along the dashed line Check work with teammate before rotating
5
Unit Circle Mastery • Station 5
Expression Arena
Table Tent
Step-by-Step Expression Strategy
1 Replace Each Trig Term: Substitute the exact numerical value of each angle first.
2 Multiply Coefficients: Compute products (e.g., \(2 \cdot \frac{1}{2} = 1\), \(4 \cdot \frac{1}{2} = 2\)).
3 Watch Double Negatives: Remember that subtracting a negative value becomes addition: \(-(-1) = +1\).
Quick Reminder:
Never mix decimal approximations into exact algebra. Evaluate fractions cleanly using integer addition.
Station Protocol: Show intermediate substitutions before giving the final integer result! 8–10 Min Limit
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Station 5 Problems: Solve on Your Team Passport
Page 4 Log
Task 5A: Linear Evaluation (Set 1)
Evaluate the exact numerical value:
\(2\cos(60^\circ) + 4\sin(30^\circ) - \tan(135^\circ)\)
• \(\cos(60^\circ) = \frac{1}{2}\)
• \(\sin(30^\circ) = \frac{1}{2}\)
• \(\tan(135^\circ) = -1\)
Task 5B: Linear Evaluation (Set 2)
Evaluate the exact numerical value:
\(4\cos(60^\circ) - 2\sin(210^\circ) + \tan(225^\circ)\)
• \(\cos(60^\circ) = \frac{1}{2}\)
• \(\sin(210^\circ) = -\frac{1}{2}\)
• \(\tan(225^\circ) = 1\)
Unit Circle Table Tent 5 • Fold in half along the dashed line Check work with teammate before rotating
6
Unit Circle Mastery • Station 6
Coordinate Mystery Detective
Table Tent
The Unit Circle Equation: \(x^2 + y^2 = 1\)
Solving for \(y\): \(y^2 = 1 - x^2 \implies y = \pm\sqrt{1 - x^2}\)
Solving for \(x\): \(x^2 = 1 - y^2 \implies x = \pm\sqrt{1 - y^2}\)
Quadrant Coordinate Signs:
QI: \((+, +)\)
QII: \((-, +)\)
QIII: \((-, -)\)
QIV: \((+, -)\)
Station Protocol: Always verify whether your solved coordinate is positive or negative! 8–10 Min Limit
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Station 6 Problems: Solve on Your Team Passport
Page 4 Log
Clue 6A: Coordinate Search in Quad II
Point \(P\left(-\frac{1}{2}, y\right)\) is on the unit circle in Quadrant II.
Solve for exact \(y > 0\) using \(x^2 + y^2 = 1\).
Find exact \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\).
In Quad II, \(x < 0\) and \(y > 0\).
Clue 6B: Coordinate Search in Quad III
Point \(Q\left(x, -\frac{1}{2}\right)\) is on the unit circle in Quadrant III.
Solve for exact \(x < 0\) using \(x^2 + y^2 = 1\).
Find exact \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\).
In Quad III, both \(x < 0\) and \(y < 0\).
Unit Circle Table Tent 6 • Fold in half along the dashed line Check work with teammate before rotating
