Circle Sleuth Slides Trigonometry Expedition
Radius = 1 • Center (0, 0)
Circle Sleuth:
The Unit Circle Scavenger Hunt
Decode the celestial coordinate grid. Convert degrees and radians, verify quadrant signs, and trace the 12-station navigator's loop.
12 Interactive Stations
25-30 Minute Search
1 Mystery Cipher Code
The Navigator's Code
Essential Trigonometric Foundations
1
Point Formula
Every terminal point on the unit circle satisfies:
\((x, y) = (\cos \theta, \sin \theta)\)
\(x\) measures horizontal distance; \(y\) measures vertical height.
2
Quadrant Signs
Track signs by quadrant position:
QI: \((+, +)\)
QII: \((-, +)\)
QIII: \((-, -)\)
QIV: \((+, -)\)
Remember: All Students Take Calculus for positive ratios!
3
Angle Families
Master the reference values:
\(30^\circ = \frac{\pi}{6}\) \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
\(45^\circ = \frac{\pi}{4}\) \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)
\(60^\circ = \frac{\pi}{3}\) \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
Quadrantal: \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\) land on the axes.
Unit Circle Rule: Radians = Degrees \(\times \frac{\pi}{180^\circ}\)
How the Scavenger Hunt Works
Self-Checking Loop Structure
01
Begin Station
Start at any open card around the room. Record its station number on your tracker.
Scatter evenly to avoid crowds.
02
Solve Riddle
Read the coordinate riddle or angle challenge. Calculate the angle and exact \((x, y)\).
Write exact radical values.
03
Log Clue
Write down the station's Secret Cipher Letter and note your calculated destination angle.
Collect all 12 cipher letters.
04
Next Leap
Search the room for the card displaying your calculated value at the top banner!
Loops back to your start!
Self-Checking Loop: If your solved coordinate cannot be found at any station, check your quadrant signs or radian conversions before asking for a lifeline!
Sample Clue Breakdown
Decode & Navigate Example
STATION 04 Cipher Letter: [ R ]
Navigator Clue:
"Find the terminal position where \(\theta = 150^\circ\). Identify its radian measure and exact coordinate pair \((x, y)\)."
→ Your answer leads to the next station banner!
A
Convert to Radians
\(150^\circ \times \frac{\pi}{180^\circ} = \frac{15\pi}{18} = \frac{5\pi}{6}\)
B
Determine Reference & Signs
Quadrant II: \(x < 0\) and \(y > 0\). Reference angle \(= 180^\circ - 150^\circ = 30^\circ\).
C
Exact Coordinates
\(\left(-\cos 30^\circ, \sin 30^\circ\right) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
Next Move: Walk to the station with the banner displaying \((-\frac{\sqrt{3}}{2}, \frac{1}{2})\)!
Expedition Ground Rules
Station Conduct & Protocol
Investigate in Pairs
Collaborate actively. Both partners must verify quadrant signs and agree on the coordinates before departing a station.
Max 2 pairs per card at a time.
Track Every Step
Write down the degree, radian fraction, and exact radical \((x,y)\) pair for each station on your Mission Sheet.
No decimals allowed! Use exact radicals.
Crack the Cipher
Collect all 12 letters in your loop order. Unscramble the letters to reveal the navigator's secret trigonometric message!
Check in with your teacher when complete.
Ready, Navigators? Head to your starting station when assigned!
Timer: 25 Minutes
Expedition Debrief
Mathematical Takeaways
Reflection 1: Symmetry Advantage
Why do you only really need to memorize the first quadrant coordinates to know all 16 points on the entire unit circle?
Key: Reflections across axes preserve numbers; only signs flip!
Reflection 2: Radian Patterns
How did denominators (\(3, 4, 6\)) tell you the exact reference angle and coordinate magnitude without converting to degrees first?
Denominator 6 → \(30^\circ\); Denominator 4 → \(45^\circ\); Denominator 3 → \(60^\circ\).
Decoded Navigator Cipher
C - O - O - R - D - I - N - A - T - I - O - N
Mission Accomplished!
Circle Sleuth Clue Cards Activity Printable Station Placards • Cut along dotted lines • Hang around room
Set 1 of 3 (Stations 01–04)
Previous Answer / Key Banner \((0, 1)\) • \(\frac{\pi}{2}\) • \(90^\circ\)
STATION 01
Cipher: [ C ]
Trig Challenge
I am located in Quadrant I . My reference angle is \(30^\circ\).
1. Convert \(30^\circ\) to exact radians.
2. Find the exact terminal coordinates \((x, y)\).
Search for your coordinates at the next card!
Previous Answer / Key Banner \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\) • \(\frac{\pi}{6}\)
STATION 02
Cipher: [ O ]
Trig Challenge
Advance counter-clockwise into Quadrant II to the angle \(\theta = 135^\circ\).
1. Convert \(135^\circ\) to simplified radian measure.
2. Determine the exact coordinates \((x, y)\) with correct signs.
Search for your coordinates at the next card!
Previous Answer / Key Banner \(\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\) • \(\frac{3\pi}{4}\)
STATION 03
Cipher: [ O ]
Trig Challenge
I am a quadrantal angle lying on the negative horizontal axis directly opposite to \(0^\circ\).
1. State this angle in degrees and radians.
2. What are the exact coordinates of \(\theta = \pi\)?
Search for your coordinates at the next card!
Previous Answer / Key Banner \((-1, 0)\) • \(\pi\) • \(180^\circ\)
STATION 04
Cipher: [ R ]
Trig Challenge
Step into Quadrant III to the angle with a reference angle of \(60^\circ\) (\(\theta = 240^\circ\)).
1. Convert \(240^\circ\) to exact radians.
2. Determine the coordinates \(\left(\cos 240^\circ, \sin 240^\circ\right)\).
Search for your coordinates at the next card!
Circle Sleuth Scavenger Hunt Placards • Page 1 of 3
Printable Station Placards • Cut along dotted lines • Hang around room
Set 2 of 3 (Stations 05–08)
Previous Answer / Key Banner \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\) • \(\frac{4\pi}{3}\)
STATION 05
Cipher: [ D ]
Trig Challenge
I am the southernmost quadrantal angle , pointing along the negative vertical y-axis (\(270^\circ\)).
1. Convert \(270^\circ\) to exact radians.
2. State the coordinates \((x, y)\) of this point.
Search for your coordinates at the next card!
Previous Answer / Key Banner \((0, -1)\) • \(\frac{3\pi}{2}\) • \(270^\circ\)
STATION 06
Cipher: [ I ]
Trig Challenge
Cross over into Quadrant IV to the standard angle \(\theta = 300^\circ\).
1. Convert \(300^\circ\) to exact radians.
2. Determine the exact coordinates \((x, y)\).
Search for your coordinates at the next card!
Previous Answer / Key Banner \(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\) • \(\frac{5\pi}{3}\)
STATION 07
Cipher: [ N ]
Trig Challenge
Still in Quadrant IV , find the angle that is exactly \(30^\circ\) shy of a full \(360^\circ\) turn (\(\theta = 330^\circ\)).
1. Convert \(330^\circ\) to exact radians.
2. Determine the exact coordinates \((x, y)\).
Search for your coordinates at the next card!
Previous Answer / Key Banner \(\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\) • \(\frac{11\pi}{6}\)
STATION 08
Cipher: [ A ]
Trig Challenge
Complete the revolution to reach the positive horizontal axis where \(\theta = 0^\circ\) or \(360^\circ\) (\(2\pi\)).
1. State the exact coordinates \((x, y)\) of \(\theta = 0\).
2. From here, prepare to rotate \(60^\circ\) counter-clockwise.
Search for your coordinates at the next card!
Circle Sleuth Scavenger Hunt Placards • Page 2 of 3
Printable Station Placards • Cut along dotted lines • Hang around room
Set 3 of 3 (Stations 09–12)
Previous Answer / Key Banner \((1, 0)\) • \(0^\circ\) • \(2\pi\)
STATION 09
Cipher: [ T ]
Trig Challenge
In Quadrant I , find the angle where \(\cos\theta = \frac{1}{2}\) and \(\sin\theta = \frac{\sqrt{3}}{2}\).
1. State this angle in degrees and radians.
2. Confirm the coordinate pair \((x, y)\).
Search for your coordinates at the next card!
Previous Answer / Key Banner \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) • \(\frac{\pi}{3}\) • \(60^\circ\)
STATION 10
Cipher: [ I ]
Trig Challenge
Travel across the y-axis into Quadrant II to the angle \(\theta = \frac{5\pi}{6}\).
1. Convert \(\frac{5\pi}{6}\) to degrees.
2. Determine the exact coordinates \((x, y)\).
Search for your coordinates at the next card!
Previous Answer / Key Banner \(\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\) • \(\frac{5\pi}{6}\)
STATION 11
Cipher: [ O ]
Trig Challenge
Descend into Quadrant III where both coordinates are equal and negative: \(x = y = -\frac{\sqrt{2}}{2}\).
1. Identify this angle in degrees (\(180^\circ + 45^\circ\)).
2. Convert this angle into exact radian measure.
Search for your coordinates at the next card!
Previous Answer / Key Banner \(\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\) • \(\frac{5\pi}{4}\)
STATION 12
Cipher: [ N ]
Trig Challenge
Close the loop! Find the northernmost quadrantal angle located on the positive y-axis (\(90^\circ\)).
1. Convert \(90^\circ\) to exact radians.
2. State the exact terminal coordinates \((x, y)\).
• This connects back to Station 01!
Circle Sleuth Scavenger Hunt Placards • Page 3 of 3
Circle Sleuth Mission Tracker Worksheet Circle Sleuth: Mission Tracker
Unit Circle Scavenger Hunt • Coordinate Verification & Cipher Log
Name:
Date:
Navigator Protocol: Start at any station around the room. Solve each card's trigonometric prompt to identify the terminal angle (degrees and radians) and exact coordinates \((x, y) = (\cos\theta, \sin\theta)\). The answer will direct you to your next station card. Record each station's secret cipher letter to decode the final word!
Unit Circle Coordinate Field \((r = 1)\) Label Key Angles & Exact Values as you discover them
+x +y (1, 0) 0°, 2π (0, 1) 90°, π/2 (-1, 0) 180°, π (0, -1) 270°, 3π/2 Q I (+, +) Q II (-, +) Q III (-, -) Q IV (+, -)
Q I: All Positive \(\cos > 0, \sin > 0\)
Q II: Sine Positive \(\cos < 0, \sin > 0\)
Q III: Tan Positive \(\cos < 0, \sin < 0\)
Q IV: Cos Positive \(\cos > 0, \sin < 0\)
Circle Sleuth Mission Tracker • Student Sheet Page 1 of 2 Turn over to log all 12 station clues and cipher letters →
12-Station Expedition Log & Decryption Log
Record in your actual loop order
Step
Station #
Angle (° & rad)
Exact Coordinate \((x, y)\)
Cipher
Scratch Work / Verification
1
2
3
4
5
6
7
8
9
10
11
12
Master Navigator Cipher Box
Unscramble the 12 station letters to reveal the password:
1. Symmetry & Quadrant Analysis: Explain why \(\cos 150^\circ = -\frac{\sqrt{3}}{2}\) while \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), but both share \(\sin = \frac{1}{2}\).
2. Radians & Fractions: How does the denominator of a simplified radian fraction (like 3, 4, or 6) immediately signal which coordinate family it belongs to?
Circle Sleuth Mission Tracker • Student Sheet Page 2 of 2 Submit completed tracker & cipher code to mission command
Circle Sleuth Teacher Guide Circle Sleuth: Teacher Guide
Unit Circle Scavenger Hunt • Facilitation, Setup & Solution Key
Precalculus / Trigonometry
Duration: 45–55 Minutes
Learning Objectives
Convert fluently between degree measures and simplified radian fractions.
Evaluate exact coordinates \((x, y) = (\cos\theta, \sin\theta)\) for all standard unit circle angles.
Apply quadrant sign conventions (\(\text{ASTC}\)) and special triangle reference angles.
Self-monitor mathematical precision through a 12-station continuous circuit loop.
Materials Needed
• 12 Station Placards (cut & hung)
• 1 Mission Tracker per student
• Circle Sleuth Slide Deck
• Clipboards & Pencils
Station Preparation & Room Logistics
1. Print & Post: Print the 3 pages of Station Placards single-sided on cardstock. Cut into 12 individual cards. Hang them around the walls or desks in random order (not sequential 1–12) so students move around the entire room.
2. Stagger Launch: Divide students into pairs. Distribute pairs evenly across different stations (e.g., Pair A at Station 01, Pair B at Station 03, Pair C at Station 07). Because the activity forms a closed loop, any station serves as a valid starting point!
3. Self-Checking Circuit: The answer calculated at the current station is displayed on the top banner of the next target station. If students cannot locate their answer, they must re-check their quadrant sign or reference angle.
4. Loop Verification: When students return to their starting station banner, their circuit is complete! They will have recorded all 12 cipher letters.
Recommended Lesson Pacing
00–08 Min • Briefing
Project slides 1–4. Review \(x = \cos\theta, y = \sin\theta\), quadrant signs, and sample clue procedure.
08–35 Min • Scavenger Hunt
Pairs solve stations and move through the room. Circulate to check radicals and quadrant sign errors.
35–45 Min • Cipher Decode
Students assemble the 12 letters, decode "COORDINATION", and complete the reflection questions.
45–50 Min • Debrief
Project slide 6. Discuss reflection answers: symmetry, denominator shortcuts, and coordinate precision.
Differentiation & Accommodations
Support / Intervention: Allow students to fill out the Page 1 Unit Circle diagram using their notes first, then take the diagram on their clipboard as an anchor guide.
On-Level / Standard: Require students to solve each station algebraically without consulting pre-filled circle charts, verifying answers via reference angles.
Extension / Advanced: Challenge early finishers to compute all 6 trigonometric values (\(\tan, \csc, \sec, \cot\)) for three of the stations they solved.
Circle Sleuth Teacher Guide • Page 1 of 2 See Page 2 for Master Station Matrix, Answer Key & Loop Circuit Map →
Master Station Solution Matrix & Key
Cipher Key: COORDINATION
Station
Top Banner Key
Station Clue Angle
Calculated Terminal Coordinates
Cipher
Next Card
01
\((0, 1) • \frac{\pi}{2}\)
\(30^\circ\) (ref \(30^\circ\), QI)
\(\frac{\pi}{6} • \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
C
→ 02
02
\(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
\(135^\circ\) (ref \(45^\circ\), QII)
\(\frac{3\pi}{4} • \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)
O
→ 03
03
\(\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)
\(180^\circ\) (neg x-axis)
\(\pi • (-1, 0)\)
O
→ 04
04
\((-1, 0) • \pi\)
\(240^\circ\) (ref \(60^\circ\), QIII)
\(\frac{4\pi}{3} • \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
R
→ 05
05
\(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
\(270^\circ\) (neg y-axis)
\(\frac{3\pi}{2} • (0, -1)\)
D
→ 06
06
\((0, -1) • \frac{3\pi}{2}\)
\(300^\circ\) (ref \(60^\circ\), QIV)
\(\frac{5\pi}{3} • \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
I
→ 07
07
\(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
\(330^\circ\) (ref \(30^\circ\), QIV)
\(\frac{11\pi}{6} • \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)
N
→ 08
08
\(\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)
\(0^\circ / 360^\circ\) (pos x-axis)
\(0 \text{ rad} • (1, 0)\)
A
→ 09
09
\((1, 0) • 0^\circ\)
\(\cos=\frac{1}{2}, \sin=\frac{\sqrt{3}}{2}\)
\(60^\circ • \frac{\pi}{3} • \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
T
→ 10
10
\(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
\(\theta = \frac{5\pi}{6}\) (QII)
\(150^\circ • \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
I
→ 11
11
\(\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
\(x = y = -\frac{\sqrt{2}}{2}\) (QIII)
\(225^\circ • \frac{5\pi}{4} • \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\)
O
→ 12
12
\(\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\)
\(90^\circ\) (pos y-axis)
\(\frac{\pi}{2} • (0, 1)\)
N
→ 01 (Loop!)
Continuous Loop Progression Diagram 12 → 1 → 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9 → 10 → 11 → 12
01 [C] → 02 [O] → 03 [O] → 04 [R] → 05 [D] → 06 [I] → 07 [N] → 08 [A] → 09 [T] → 10 [I] → 11 [O] → 12 [N]
Common Misconceptions & Teacher Interventions
1. Confusing \(x\) and \(y\) for \(30^\circ\) vs \(60^\circ\): Students frequently invert \(\frac{1}{2}\) and \(\frac{\sqrt{3}}{2}\). Prompt: "Look at the unit circle diagram: is the horizontal distance wider or taller? \(30^\circ\) is wider (\(x = \frac{\sqrt{3}}{2} \approx 0.866\)) and short in height (\(y = \frac{1}{2} = 0.5\))."
2. Forgetting Negative Signs: Students often find the correct reference angle values but forget that \(x < 0\) in QII and both are negative in QIII. Have them point to the quadrant physically to verify signs before writing.
3. Quadrantal Coordinates: At \(\pi\) (\(180^\circ\)) and \(\frac{3\pi}{2}\) (\(270^\circ\)), students occasionally write \((0, -1)\) for \(180^\circ\) instead of \((-1, 0)\). Remind them that \(x\) is horizontal and \(y\) is vertical.
4. Reflection Prompt Solutions:
Q1 Solution: \(150^\circ\) is the reflection of \(30^\circ\) across the y-axis; the horizontal coordinate flips sign, while vertical height is preserved.Q2 Solution: Denominators indicate reference families: \(/6 \to 30^\circ\), \(/4 \to 45^\circ\), \(/3 \to 60^\circ\).
Circle Sleuth Teacher Guide • Page 2 of 2 Complete Key • Master Cipher: COORDINATION
