Trigonometric Identities Test – Precalculus Math (Printable PDF)

Question 1 Simplify Expressions (2 points)

Simplify. (a) cos x(sec x − cos x) (1 point) (b) (1 + tan² x) cos² x (1 point)

Question 2 Find Cosine and Tangent (2 points)

Angle θ lies in Quadrant III and sin θ = −8/17. Find cos θ and tan θ.

Question 3 Verify an Identity (3 points)

Verify that sin x1 − cos x = 1 + cos xsin x.

Question 4 Exact Values from Sum Formulas (3 points)

(a) Find the exact value of cos 105°, using 105° = 60° + 45°. (2 points) (b) Find the exact value of tan(π/12), using π/12 = π/3 − π/4. (1 point)

Question 5 Sum of Two Angles from a Graph (3 points)

The unit circle shows point P at angle α and point Q at angle β.

1. Read sin and cos of α and β from the coordinates. (1 point)
2. Find sin(α + β). (1 point)
3. Find cos(α + β). (1 point)

Question 6 Double-Angle Values (2 points)

Angle θ lies in Quadrant II and sin θ = 4/5. Find sin 2θ and cos 2θ.

Question 7 Verify with Double Angles (3 points)

Verify that sin 2x1 + cos 2x = tan x.

Question 8 Solve a Double-Angle Equation (2 points)

Solve cos 2x = sin x on [0, 2π). Use cos 2x = 1 − 2sin² x.