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Trig Review Practice Problems

Unit circle, radians, identities, and equations — with solutions.

Trigonometry is the part of precalculus that calculus assumes most aggressively. Derivatives of sin and cos, the special limit sin(x)/x → 1, trig substitution — all of it presumes you can produce exact values instantly and in radians. (Calculus is radians-only: the clean derivative rules are false in degrees.)

A working trig review comes down to four skills:

The unit circle, cold. Exact sine and cosine values for the special angles — π/6, π/4, π/3, π/2 and their multiples — without counting on your fingers.
Radians and reference angles. Convert fluently between degrees and radians, and reduce any angle to its reference angle plus a quadrant sign.
The core identities. The Pythagorean identity, quotient and reciprocal identities, and cofunction relationships — enough to rewrite any trig expression in terms of sine and cosine.
Inverse trig and equations. Know the restricted ranges of arcsin and arccos, and find all solutions of a basic equation on [0, 2π) — not just the calculator's one.

The eight problems below hit each skill in that order. No calculator — the whole point is exact values. Every answer includes a short worked solution.

Practice problems

Problem 1

Evaluate

sin (\frac{π}{4})

Problem 2

Convert

42 0^{\circ}

to radians

Problem 3

An angle of

330

(in degrees) lies in Quadrant IV. What is its reference angle?

Problem 4

Evaluate

cos (\frac{11 π}{6})

Problem 5

Evaluate

cos (\frac{π}{2} - \frac{π}{6})

Problem 6

Evaluate

arcsin (\frac{1}{2})

Problem 7

Given

cot θ = 1

and

θ

in Quadrant I, find

csc θ

Problem 8

Solve

cos x = \frac{1}{2}

for all

x \in [0, 2 π)

Want more than 8 problems?

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