Circle Navigator Teacher Guide Intervention Guide: The Infinite Circle
Topic: Unit Circle Extension (HS.F-TF.A.2)
Tier 2 Intervention
Instructional Focus
This lesson targets students who struggle to transition from "Right Triangle Trig" (SOH CAH TOA) to "Unit Circle Trig." The goal is for students to see sine and cosine as coordinates on a plane, rather than just ratios in a triangle.
Standards Alignment
CO HS.F-TF.A.2
"Explain how the unit circle... enables the extension of trig functions to all real numbers."
Lesson Flow
1
The "Coordinate Shift" (10 mins)
Transition students from \( \text{Opp}/\text{Hyp} \) to \( y/r \). On a unit circle, \( r=1 \), so \( \sin \theta = y \) and \( \cos \theta = x \).
Prompt: "In Quadrant I, both x and y are positive. What happens when we move to Quadrant II?"Key Check: Do students recognize that \(x\) becomes negative?
2
Scaffolded Exploration (15 mins)
Use the Coordinate Navigator Worksheet . Students use the "Finger-Trace" method to move counter-clockwise from 0°.
Students should identify the sign (+/-) of the coordinate before worrying about the value.
Introduce ASTC (All Students Take Calculus) as a mnemonic for where functions are positive.
3
Progress Monitoring (5 mins)
Administer the Quadrant Check . This provides instant data on whether students can extend the concept to different quadrants.
Common Stumbling Blocks
X/Y Confusion
Students often swap sine and cosine (thinking sine is x).
Fix: Remind them that Alphabetically, (C, S) matches (x, y).
Reference Angle Error
Measuring from the y-axis instead of the x-axis.
Fix: Always draw the "bow-tie" back to the x-axis.
Quick Answer Key
Q1
\(x: +, y: +\)
\( \cos: +, \sin: + \)
Q2
\(x: -, y: +\)
\( \cos: -, \sin: + \)
Q3
\(x: -, y: -\)
\( \cos: -, \sin: - \)
Q4
\(x: +, y: -\)
\( \cos: +, \sin: - \)
Infinite Circle Slides THE INFINITE CIRCLE
Trigonometry Beyond Right Triangles
Making the Shift
Before: SOH CAH TOA
\( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
\( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
Limited to angles between 0° and 90°.
Now: The Unit Circle
\( \sin \theta = y \)
\( \cos \theta = x \)
Works for EVERY angle in existence.
The Coordinate Connection
P(x, y) θ
Every point on the circle has an (x, y) coordinate.
X is the Cosine of the angle
Y is the Sine of the angle
The Map of Signs
QUADRANT II
(- , +)
Cos: Negative
Sin: Positive
Quadrant I
(+ , +)
Cos: Positive
Sin: Positive
QUADRANT III
(- , -)
Cos: Negative
Sin: Negative
QUADRANT IV
(+ , -)
Cos: Positive
Sin: Negative
WHO IS POSITIVE?
A
All
Quad I: Everyone is +
S
Students
Quad II: Only Sine is +
T
Take
Quad III: Only Tan is +
C
Calculus
Quad IV: Only Cos is +
Circle Navigator Worksheet Circle Navigator
Extension of Sine and Cosine to All Real Numbers
Student:
Date:
PART 1: The Coordinate Switch
On a unit circle (radius = 1), the coordinates of any point \(P\) are defined by the angle \( \theta \).
x
Coordinate \(x\) =
y
Coordinate \(y\) =
(x, y)
PART 2: Sign Hunter
Identify if the value is Positive (+) or Negative (-) based on the quadrant.
Angle: 150° Quadrant II
\( \cos(150^\circ) \)
-
\( \sin(150^\circ) \)
-
Angle: 225° Quadrant III
\( \cos(225^\circ) \)
-
\( \sin(225^\circ) \)
-
Angle: 330° Quadrant IV
\( \cos(330^\circ) \)
-
\( \sin(330^\circ) \)
-
Angle: 120° Quadrant II
\( \cos(120^\circ) \)
-
\( \sin(120^\circ) \)
-
PART 3: Beyond 90 Degrees
Step-by-step: Finding the exact value for \( \sin(210^\circ) \).
Step 1: Sketch it
Draw the terminal side at 210°
Step 2: Reference Angle
How many degrees is the angle away from the x-axis (180°)?
Step 3: Combine Sign + Value
Quadrant III sine is _________. Reference angle \( \sin(30^\circ) = 1/2 \).
Final Answer: \( \sin(210^\circ) = \)
Quadrant Quest Check Quadrant Quest Check
Progress Monitoring: HS.F-TF.A.2
Name
1 Where does it land?
For each angle, identify the Quadrant (I, II, III, or IV).
230°
110°
325°
2 Sign Hunter
Circle the correct sign for each function.
\( \sin(190^\circ) \)
-
\( \cos(305^\circ) \)
-
\( \cos(95^\circ) \)
-
3 The Big Picture
Explain why the Unit Circle allows us to find the cosine of 200°, even though we can't draw a right triangle with a 200° angle.
Self-Reflection:
Lost
Getting There
Mastered It
