Computing Trigonometric Values
Evaluating Trigonometric Expressions — SAT Math Mastery
What Is Evaluation?
Evaluating a trigonometric expression means finding its numerical value. A trigonometric expression is any expression containing $\sin$, $\cos$, $\tan$, or their reciprocals ($\csc$, $\sec$, $\cot$).
Two types of answers:
- Exact values — written with radicals and fractions, e.g. $\dfrac{\sqrt{3}}{2}$. Use when the angle is a special angle or answer choices are in radical form.
- Decimal approximations — e.g. $0.866$. Use when a calculator is needed or the problem asks you to round.
Why this skill matters: It is the foundation for solving trig equations, graphing trig functions, and applying trigonometry to real-world problems. The SAT tests this directly and indirectly.
📐 Review: Trigonometric Ratios (SOH-CAH-TOA)
Primary Ratios
| Function | Ratio |
|---|---|
| $\sin(\theta)$ | $\dfrac{\text{opposite}}{\text{hypotenuse}}$ |
| $\cos(\theta)$ | $\dfrac{\text{adjacent}}{\text{hypotenuse}}$ |
| $\tan(\theta)$ | $\dfrac{\text{opposite}}{\text{adjacent}}$ |
Quotient relationship: $\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}$
Reciprocal Functions
| Function | Definition | Ratio |
|---|---|---|
| $\csc(\theta)$ | $\dfrac{1}{\sin(\theta)}$ | $\dfrac{\text{hyp}}{\text{opp}}$ |
| $\sec(\theta)$ | $\dfrac{1}{\cos(\theta)}$ | $\dfrac{\text{hyp}}{\text{adj}}$ |
| $\cot(\theta)$ | $\dfrac{1}{\tan(\theta)}$ | $\dfrac{\text{adj}}{\text{opp}}$ |
