Trigonometry Trigonometric Identities 3 — Questions and Answers
Question 1: Which of the following is the correct half-angle formula for sin(θ/2)?
- ±√((1 − cos θ)/2) (Correct answer)
- ±√((1 + cos θ)/2)
- ±√(2/(1 − cos θ))
- ±√(cos θ/2)
Correct answer: ±√((1 − cos θ)/2)
The half-angle identity is sin(θ/2) = ±√((1 − cos θ)/2), with the sign determined by the quadrant of θ/2.
Question 2: Simplify: tan θ · cos θ
- sin θ (Correct answer)
- sec θ
- cot θ
- csc θ
Correct answer: sin θ
tan θ = sin θ/cos θ, so tan θ · cos θ = (sin θ/cos θ) · cos θ = sin θ.
Question 3: Which identity represents the product-to-sum formula for sin A · cos B?
- (1/2)[sin(A+B) + sin(A−B)] (Correct answer)
- (1/2)[cos(A−B) − cos(A+B)]
- (1/2)[cos(A+B) + cos(A−B)]
- (1/2)[sin(A+B) − sin(A−B)]
Correct answer: (1/2)[sin(A+B) + sin(A−B)]
The product-to-sum formula gives sin A cos B = (1/2)[sin(A+B) + sin(A−B)].
Question 4: What is sin(3π/2) + cos(π)?
- −2 (Correct answer)
- 0
- 2
- −1
Correct answer: −2
sin(3π/2) = −1 and cos(π) = −1, so their sum is −1 + (−1) = −2.
Question 5: If sin x = cos x, which identity confirms the solution x = π/4 + nπ?
- Dividing both sides by cos x gives tan x = 1 (Correct answer)
- Using sin²x + cos²x = 1
- Using the double-angle formula
- Using the half-angle formula
Correct answer: Dividing both sides by cos x gives tan x = 1
Dividing sin x = cos x by cos x yields tan x = 1, which holds when x = π/4 + nπ.
Question 6: Which of the following is an even function identity?
- cos(−θ) = cos θ (Correct answer)
- sin(−θ) = sin θ
- tan(−θ) = tan θ
- sec(−θ) = −sec θ
Correct answer: cos(−θ) = cos θ
Cosine is an even function, so cos(−θ) = cos θ; sine and tangent are odd functions.
Question 7: What does the identity sin(π − θ) simplify to?
- sin θ (Correct answer)
- −sin θ
- cos θ
- −cos θ
Correct answer: sin θ
Using the sine difference formula: sin(π − θ) = sin π cos θ − cos π sin θ = 0 − (−1)sin θ = sin θ.
Trigonometry Trigonometric Identities 3
Which of the following is the correct half-angle formula for sin(θ/2)?