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. Use the identities to find values and simplify expressions
2. Explain why for any angle
3. Rearrange it; choose the sign from the quadrant
4. Derive and

Unit Circle Creates the Fundamental Identity

gives for every angle

Squares matter: at , , not 1.

Rearranged Forms Isolate One Function

Subtract to rearrange:

Take the root: . The quadrant picks the sign (ASTC).

Worked Example: Same Value, Two Quadrants

If , find .

QI: cosine positive,
QII: cosine 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

We have one identity: can we get more? Divide every term by :

Dividing by Sine Squared Gives Identity Three

Divide every term of by :

The Identity Family as One System

Quotient identities (bottom row) are recalled, not re-taught.

Don't confuse (two functions) with (one).

Worked Example: Find All Six Values

If and is in QI, find all six trig values.

, so (QI: positive)

,

,

Quick Check: Use the Derived Identity

If and is in QI, find .

Which identity links cot and csc? Then choose the sign...

Strategy: Spot the Identity, Then Simplify

1. Look for , , , or
2. Substitute the matching identity
3. If a reciprocal or is left, write it in and , then cancel
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: Substitute, Then Cancel

Simplify and

Quick Check: Spot the Identity, Then Simplify

Simplify .

Spot the pattern, substitute, then cancel...

Worked Example: Find Values With Quadrant Analysis

If and is in QIII, find and .

QIII: sine is negative, so

60-Second Challenge: Name the Identity

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
• Spot the identity, substitute, then cancel

Watch out:

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

Coming Up Next in Trigonometry

Up next: Double-angle and sum/difference formulas

• Build on today's five identities
• Double-angle formulas such as
• Sum and difference formulas such as