Introduction
You already know the Pythagorean Theorem: . When we place a right triangle on the unit circle, a fascinating thing happens: the sides become , , and .
Prerequisite Connection
You know the Pythagorean Theorem and the definitions of the six trig functions on the unit circle.
Today's Increment
We're deriving three Pythagorean Identities that relate the six trig functions—the foundation for all identity work.
Why This Matters
In Calculus, you'll use these identities constantly to simplify integrals. For example, requires rewriting using an identity.
The Three Pythagorean Identities
Identity #1
Identity #2
Identity #3
Derivation (Identity #2)
Start with Identity #1 and divide every term by :
This simplifies to:
Worked Examples
Example 1: Simplifying an Expression
Simplify .
Apply Identity #1
.
Substitute
Expression becomes .
Apply Identity #2
.
Answer:
Example 2: Finding Exact Values
If and is in Quadrant I, find .
Use Identity #1
Substitute Known Value
Solve for Cosine
Since QI, .
Answer:
Example 3: Proving an Identity (Advanced)
Show that .
Recognize Identity #2
We know .
Rearrange Algebraically
Subtract from both sides:
QED. The identity is proven. ✓
Common Pitfalls
Forgetting the ± when taking square roots
If , then . The quadrant determines the sign.
Confusing the identities
It's , NOT . The "1" is always on the side with the squared cotangent or tangent.
Real-World Application
Signal Processing & Electrical Engineering
When analyzing alternating current (AC) circuits, engineers represent voltage and current as sinusoidal functions. The Pythagorean identity is used to calculate the power factor and to convert between real power, reactive power, and apparent power in the "power triangle."
Practice Quiz
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