Prove and Use the Pythagorean Identity

In this lesson:

• Prove from the unit circle
• Find missing trig values using the identity and quadrant
• Derive the related identities for tangent and secant

Learning Objectives for This Lesson

By the end, you will be able to:

1. Prove the Pythagorean identity from the unit circle
2. Explain why the identity holds for all angles
3. Find a missing trig value using the identity and quadrant
4. Derive the related forms for secant and cosecant

What You Already Know About Circles

The unit circle has radius 1, centered at the origin.

• Every point satisfies
• For angle : the point is
• By definition: and

Substitute those definitions — what do you get?

Setting Up the Pythagorean Trig Identity

On the unit circle, the point for angle has coordinates .

Substitute the definitions of sine and cosine:

Visual Evidence for the Pythagorean Identity

The right triangle inside the unit circle has legs and — hypotenuse is always 1.

Verifying the Identity at Specific Angles

Check-In: Why Does It Work for All Angles?

Claim: The identity holds for , , even radian.

Why does the proof work for these angles, not just for 30-60-90 triangles?

Think about it before the next slide...

Why the Identity Holds for All Angles

The proof uses one fact: every point on the unit circle satisfies .

• Every angle gives a point on the circle
• Therefore: for every

This is a theorem — not a pattern from examples.

Two-Step Process: Using the Identity

Given one trig value and a quadrant, find the other in two steps:

1. Identity gives magnitude — solve for the squared value, take root
2. Quadrant gives sign — select or from quadrant rules

Both steps are required — skipping step 2 gives an incomplete answer.

Reference Chart for Quadrant Sign Rules

Memory aid: All Students Take Calculus (QI, QII, QIII, QIV)

Example 1: Given Cosine, Find Sine (QI)

Given: , in QI. Find .

Step 1: Apply the identity:

Step 2: Take the square root:

Step 3: Apply the quadrant — QI means :

Example 2: Given Sine, Find Cosine (QIII)

Given: , in QIII. Find .

Step 1:

Step 2:

Step 3: QIII means :

Example 3: Irrational Result (QII)

Given: , in QII. Find .

Step 1:

Step 2:

Step 3: QII means :

Your Turn: Guided Identity Practice

Find given , in QII.

Step 1: Solve for using the identity

Step 2: Take the square root — write

Step 3: Select the sign for QII

Work through all three steps before advancing

Practice: Finding Missing Trig Values

Find the missing value for each:

1. , in QIV — find
2. , in QIII — find
3. , in QII — find

Pause and solve all three before advancing

Answers to Pythagorean Identity Practice

1. (QIV: )

2. (QIII: )

3. (QII: )

Using the Identity to Find Tangent

Once and are known, tangent is their ratio:

• QI/QIII: same signs →
• QII/QIV: opposite signs →

Example: Extend Example 1 to Find Tangent

From Example 1: , , in QI

From Example 2: , , in QIII

Finding Tangent Directly from Identity

Given: , in QIII. Find directly.

Step 1: Find using the identity:

Step 2: Compute tangent:

Quick Check: Tangent in QII

If and is in QII, what is ?

Work through the three steps:

1. Find from the identity
2. Apply the quadrant sign for QII
3. Compute

Check your answer before continuing

Derived Identity 1: Divide by

Start with:

Divide every term by (assuming ):

Derived Identity 2: Divide by

Start with:

Divide every term by (assuming ):

All Three Pythagorean Identity Forms

All three come from the same source — just divided by different values.

Example: Using the Derived Identity

Given: , QII. Find and .

Step 1: :

Step 2: , so

Step 3: QII →

Practice Problems: Derived Identity Forms

Solve each:

1. , in QIV — find
2. , in QII — find
3. Simplify: — what does it equal?

Pause and solve before advancing

Answers to Derived Identities Practice

1. QIV: →

2. QII: →

3. — by definition

Key Takeaways from This Lesson

• holds for every angle
• Identity gives ; quadrant determines the final sign
• Divide by or for derived forms
• — the identity uses squared terms
• , not

What's Next: Angle Addition Formulas

Next lesson: HSF.TF.C.9 — Angle Addition and Subtraction Formulas

The Pythagorean identity appears in the proofs of these formulas.