Trigonometric Identities
SAT Math — Complete Master Guide
Category: Trigonometry • SAT Reasoning Test
1 — Introduction: What Are Trigonometric Identities?
What is an Identity?
An identity is an equation that is true for every value of the variable for which both sides are defined. Compare:
| Type | Example | True for… |
|---|---|---|
| Identity | $\sin^2\theta + \cos^2\theta = 1$ | All $\theta$ |
| Equation | $\sin\theta = \tfrac{1}{2}$ | Only certain $\theta$ |
Why Identities Matter on the SAT
- Simplify complicated expressions into simpler ones.
- Solve trigonometric equations by rewriting in one function.
- Find exact values of non-special angles (e.g., $75°$).
- Verify that two expressions are equivalent.
Verifying ≠ Solving. When you verify an identity you show both sides are always equal. When you solve an equation you find specific angle(s). On the SAT, most questions ask you to simplify or find a value — not write a formal proof.
Strategy Overview
- Pick the more complicated side.
- Convert everything to $\sin$ and $\cos$.
- Use Pythagorean / reciprocal / quotient identities to simplify.
- Factor, combine fractions, or multiply by a conjugate as needed.
- Arrive at the other side.
The Unit Circle — Foundation of All Identities
Every point on the unit circle is $(\cos\theta,\,\sin\theta)$, and $\cos^2\theta+\sin^2\theta=1$ follows directly from $x^2+y^2=1$.
Interactive visualization — available in the full lesson
Drag the slider to change $\theta$. Notice $\sin^2\theta+\cos^2\theta$ always equals 1.
2 — Fundamental Definitions (SOH-CAH-TOA)
| Function | Definition | Mnemonic |
|---|---|---|
| $\sin\theta$ | $\dfrac{\text{opposite}}{\text{hypotenuse}}$ | SOH |
| $\cos\theta$ | $\dfrac{\text{adjacent}}{\text{hypotenuse}}$ | CAH |
| $\tan\theta$ | $\dfrac{\text{opposite}}{\text{adjacent}}$ | TOA |
These three ratios are the foundation for every identity that follows.