Trigonometric Identities Test Answer Key with Scoring – Precalculus

Question 1 Simplify Expressions (2 points)

(a) 1 − cos² x = sin² x.

(b) sec² x · cos² x = 1.

Scoring: 1 point per part.

Question 2 Find Cosine and Tangent (2 points)

cos² θ = 1 − 64/289 = 225/289. In Quadrant III cosine is negative: cos θ = −15/17.

tan θ = (−8/17) ÷ (−15/17) = 8/15.

Scoring: 1 point per value.

Question 3 Verify an Identity (3 points)

Multiply the numerator and denominator of the left side by (1 + cos x): sin x(1 + cos x)/(1 − cos² x).

Since 1 − cos² x = sin² x, this equals sin x(1 + cos x)/sin² x = (1 + cos x)/sin x. ∎

Scoring: 1 point for a valid first step on one side; 1 point for using the Pythagorean identity; 1 point for reaching the other side.

Question 4 Exact Values from Sum Formulas (3 points)

(a) cos(60° + 45°) = cos 60° cos 45° − sin 60° sin 45° = (1/2)(√2/2) − (√3/2)(√2/2) = (√2 − √6)/4.

(b) tan(π/3 − π/4) = (√3 − 1)/(1 + √3) = (√3 − 1)²/(3 − 1) = (4 − 2√3)/2 = 2 − √3.

Scoring: (a) 1 point for the correct formula and values, 1 point for the simplified answer. (b) 1 point.

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

1. cos α = 12/13, sin α = 5/13; cos β = −4/5, sin β = 3/5.
2. sin(α + β) = (5/13)(−4/5) + (12/13)(3/5) = (−20 + 36)/65 = 16/65.
3. cos(α + β) = (12/13)(−4/5) − (5/13)(3/5) = (−48 − 15)/65 = −63/65.

Scoring: 1 point per part.

Question 6 Double-Angle Values (2 points)

cos θ = −3/5.

sin 2θ = 2(4/5)(−3/5) = −24/25.

cos 2θ = 9/25 − 16/25 = −7/25.

Scoring: 1 point per value.

Question 7 Verify with Double Angles (3 points)

Use sin 2x = 2 sin x cos x and 1 + cos 2x = 1 + 2cos² x − 1 = 2cos² x.

The fraction is 2 sin x cos x/(2cos² x) = sin x/cos x = tan x. ∎

Scoring: 1 point for the numerator; 1 point for the denominator; 1 point for simplifying to tan x.

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

1 − 2sin² x = sin x, so 2sin² x + sin x − 1 = 0 and (2 sin x − 1)(sin x + 1) = 0.

sin x = 1/2 gives π/6, 5π/6; sin x = −1 gives 3π/2. x = π/6, 5π/6, 3π/2.

Scoring: 1 point for the quadratic in sine; 1 point for all three solutions.