TMUA Trigonometry 3 — Questions and Answers
Question 1: Which expression is equivalent to cos(2x)?
- 1 - 2sin²(x) (Correct answer)
- 2sin(x)cos(x)
- cos²(x) - 1
- 2cos(x) - 1
Correct answer: 1 - 2sin²(x)
cos(2x) = 1 - 2sin²(x) is one of the standard double-angle identities.
Question 2: Solve for x in [0, 2π]: 2cos(x) - 1 = 0.
- x = π/3 or x = 5π/3 (Correct answer)
- x = π/6 or x = 5π/6
- x = π/4 or x = 7π/4
- x = π/3 only
Correct answer: x = π/3 or x = 5π/3
cos(x) = 1/2 gives x = π/3 and x = 5π/3 in [0, 2π].
Question 3: What is tan(π/4 + π/4) using the addition formula?
- undefined (Correct answer)
- 1
- 0
- 2
Correct answer: undefined
tan(π/2) is undefined because it equals tan(π/4 + π/4) and cos(π/2) = 0.
Question 4: Express sin(75°) using the sum formula.
- (√6 + √2)/4 (Correct answer)
- (√6 - √2)/4
- (√2 + 1)/4
- √3/2
Correct answer: (√6 + √2)/4
sin(75°) = sin(45° + 30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4.
Question 5: For which values of x is the equation sin²(x) = sin(x) satisfied in [0, 2π]?
- x = 0, π/2, π, 2π (Correct answer)
- x = 0, π/2
- x = π/2, 3π/2
- x = 0, π
Correct answer: x = 0, π/2, π, 2π
sin²x = sin x ⟹ sin x(sin x - 1) = 0 ⟹ sin x = 0 or sin x = 1, giving x = 0, π, 2π, π/2.
Question 6: The function f(x) = sin(x) + cos(x) can be written in the form R·sin(x + φ). What is R?
- √2 (Correct answer)
- 2
- 1
- √3
Correct answer: √2
R = √(1² + 1²) = √2 for f(x) = sin(x) + cos(x).
Question 7: What is the exact value of sin(π/12)?
- (√6 - √2)/4 (Correct answer)
- (√6 + √2)/4
- √3/2
- 1/2
Correct answer: (√6 - √2)/4
sin(π/12) = sin(15°) = sin(45° - 30°) = (√6 - √2)/4.
TMUA Trigonometry 3
Which expression is equivalent to cos(2x)?