Pythagorean Trig Identities

Trigonometry — Unit Circle to ACT Mastery

In this lesson:

• Derive and apply
• Use derived and quotient identities
• Simplify expressions for the ACT

What You Will Learn Today

1. Derive from the unit circle
2. Rearrange to find one trig value from another
3. Apply and
4. Use quotient identities to rewrite expressions
5. Simplify trig expressions for the ACT

Unit Circle Creates the Fundamental Identity

Every point on the unit circle satisfies

The Pythagorean Identity From the Circle

Since any point on the unit circle is :

This works for every angle — not just acute angles from right triangles.

Rearranged Forms Isolate One Function

Start with and subtract:

Key: Always subtract to rearrange — the result must be .

Worked Example: Find Cosine in Quadrant One

If and is in QI, find .

Step 1: Apply the identity

Step 2: Choose the sign using the quadrant

Same Values but Quadrant Two Changes Sign

If and is in QII, find .

Step 1: Same computation

Step 2: Quadrant II — cosine is negative

Quick Check: Find the Missing Value

If and is in QI, find .

Apply the rearranged identity, then choose the sign...

Dividing Gives Us a Second Identity

Divide every term of by :

Rule: Divide every term — not just some.

Dividing by Sine Squared Gives Identity Three

Divide every term of by :

Three Pythagorean Identities as a Family

One identity generates all three — divide to derive.

Quotient Identities Connect Trig Functions

The quotient identities express tangent and cotangent in terms of sine and cosine:

Don't confuse: (two functions) vs. (one function)

Worked Example: Find Secant From Tangent

If and is in QI, find .

Step 1: Apply the derived identity

Step 2: Solve and choose sign

Quick Check: Simplify This Expression

Simplify

Hint: rewrite secant using a reciprocal identity...

ACT Strategy: Convert Everything to Sine and Cosine

When simplifying trig expressions on the ACT:

1. Replace , , , with and
2. Simplify fractions and cancel common factors
3. Look for Pythagorean identity patterns
4. Match to the answer choices

Worked Example: Chain Two Identities Together

Simplify

Step 1: Spot the Pythagorean identity

Step 2: Substitute and recognize identity two

Worked Example: Find Values With Quadrant Analysis

If and is in QIII, find .

Step 1: Apply the identity

Step 2: Choose the sign — QIII means sine is negative

Quick Check: Match Expression to Simplified Form

Which identity simplifies each expression?

Identify the matching identity for each row...

Complete Identity Reference Card for ACT

No formula sheet on the ACT — memorize all five.

Key Takeaways and Common Mistakes

• — the foundation
• Divide to derive and
• Convert to and to simplify

Watch out:

• , not — squares matter
• Quadrant determines the sign — use ASTC

Coming Up Next in Trigonometry

Up next: ACT timed identity practice

• Apply all five identities under time pressure
• Multi-step simplification problems at the 33-36 level
• Sixty-second pacing drills for test day