Extending Trig Functions via Unit Circle

HSF.TF.A.2

In this lesson:

• Redefine sine and cosine as coordinates on the unit circle
• Evaluate trig functions in all four quadrants
• Extend definitions to negative angles and angles

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Define sine and cosine as unit circle coordinates
2. Evaluate sine and cosine in all four quadrants
3. Determine signs by quadrant
4. Extend to negative and large angles
5. Use reference angles to evaluate

Right Triangles Cannot Handle Obtuse Angles

In right-triangle trig, sine = opposite ÷ hypotenuse.

Problem: What is ?

• 135° is obtuse — no right triangle has a 135° angle
• Right-triangle definitions break down for angles ≥ 90°

We need a new definition. The unit circle provides it.

Setting Up the Unit Circle Definition

The unit circle: center at the origin, radius .

For any angle , the terminal side meets the unit circle at point .

Definitions:

Right Triangle to Coordinates (First Quadrant)

For a QI angle: hypotenuse , adjacent , opposite .

The Coordinate Definition Works for Any Angle

For angle in any quadrant: find point on the unit circle.

Example: At , the terminal side hits .

Evaluate at the Axis Angles

Quick Check: Evaluating at Axis Angles

What are and ?

State the coordinates of the point at on the unit circle.

Determining Signs in Each Quadrant

Signs follow from coordinate geometry — not rules to memorize.

Reference Chart for Quadrant Signs

Mnemonic: All Students Take Calculus (All, Sin, Tan, Cos positive)

"All Students Take Calculus" — Grounded in Coordinates

Worked: Evaluate and

Step 1: is between and → QII

Step 2: Find the point. (from unit circle)

Worked: Evaluate and

Step 1: is between and → QIII

Step 2: In QIII, both coordinates are negative.

Your Turn: Evaluate a Third Quadrant Angle

Steps to follow:

1. Identify the quadrant (compare to , , , )
2. Determine the sign of in that quadrant
3. Find the -coordinate of the unit circle point

Try it, then advance.

Quick Check: Applying Quadrant Sign Rules

In which quadrant is and ?

Use the coordinate sign chart — which quadrant has positive and negative ?

Negative Angles Always Rotate Clockwise

By convention, positive angles rotate counterclockwise.

Negative angles rotate clockwise:

• means rotate clockwise from

This lands in QIV, at the same point as .

Coterminal Angles Share Trig Values

Coterminal angles differ by multiples of and have the same terminal side.

Example: and are coterminal.

Same terminal point → same and values.

Negative and Large Angles Diagram

Every real number maps to exactly one point on the circle.

Worked Example: Evaluating a Negative Angle

Step 1: Clockwise rotation of from → QIV

Step 2: is coterminal with

Step 3: At , the point is

Worked Example: Reducing a Large Angle

Step 1: Find a coterminal angle in :

Step 2: is in QIII; point is

Your Turn: Negative Angle Evaluation

Steps:

1. Find the coterminal angle in (add )
2. Identify the quadrant
3. Read the -coordinate

Try all three steps, then advance.

Quick Check: Negative Angle Signs

Is positive or negative? Explain why.

Hint: which quadrant does land in?

Reference Angles: The Key Strategy

The reference angle is the acute angle between the terminal side and the x-axis.

• Always between 0 and
• Always positive — it is an angle size, not a directed angle

The trig values of equal the trig values of , up to sign.

Reference Angle Formulas by Quadrant

Reference Angle in Each Quadrant

Reference angle is always the angle to the x-axis, not from the positive x-axis.

Worked: Find Using Reference Angle

Step 1: is in QII (between and )

Step 2: Reference angle:

Step 3:

Step 4: In QII, → answer is positive

Worked: Find — Four Steps

Step 1: is in QIII (between and )

Step 2: Reference angle:

Step 3:

Step 4: In QIII, → answer is negative

Your Turn: Find a Fourth Quadrant Value

Apply the four-step reference angle strategy:

1. Identify the quadrant
2. Compute the reference angle using the correct formula
3. Evaluate of the reference angle
4. Apply the correct sign from the quadrant

Complete all four steps, then advance.

Practice: Three Unit Circle Evaluations

Evaluate using the reference angle strategy:

3. (find coterminal angle first)

Write out all four steps for each.

Key Takeaways from This Lesson

• , on the unit circle
• Signs: QI, QII, QIII, QIV
• Negative angles go clockwise; coterminal angles share values
• Evaluate: quadrant → ref angle → value → sign

Watch out: negative input ≠ negative output; ref angle is to x-axis

Next Steps: Exact Values from Special Triangles

Coming up — HSF.TF.A.3:

• Derive exact values of sin, cos, tan at , ,
• Use unit circle symmetry to extend values to all quadrants
• Build the complete exact-value reference table