PRE CALCULUS Trigonometry 3 — Questions and Answers
Question 1: Using the sum formula, what is the exact value of sin(75°)?
- (√6 + √2)/4 (Correct answer)
- (√6 - √2)/4
- (√3 + 1)/4
- (√3 - 1)/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 2: The Law of Cosines, c² = a² + b² - 2ab·cos(C), reduces to the Pythagorean theorem when:
- C = 90° (Correct answer)
- C = 0°
- C = 180°
- a = b
Correct answer: C = 90°
When C = 90°, cos(90°) = 0, so the -2ab·cos(C) term vanishes and c² = a² + b².
Question 3: What is the reference angle for 210°?
- 30° (Correct answer)
- 60°
- 150°
- 45°
Correct answer: 30°
210° is in the third quadrant; its reference angle is 210° - 180° = 30°.
Question 4: Which equation represents the inverse relationship arcsin(x)?
- y = arcsin(x) means sin(y) = x for -π/2 ≤ y ≤ π/2 (Correct answer)
- y = arcsin(x) means sin(x) = y for 0 ≤ y ≤ π
- y = arcsin(x) means sin(y) = x for 0 ≤ y ≤ π
- y = arcsin(x) means sin(x) = y for -π/2 ≤ x ≤ π/2
Correct answer: y = arcsin(x) means sin(y) = x for -π/2 ≤ y ≤ π/2
arcsin(x) = y means sin(y) = x, and the range is restricted to [-π/2, π/2] to make it a function.
Question 5: In a triangle, if A = 30°, a = 5, and b = 8, the Law of Sines gives sin(B) equal to:
- 4/5 (Correct answer)
- 5/8
- 8/5
- 1/2
Correct answer: 4/5
sin(B)/b = sin(A)/a → sin(B) = b·sin(A)/a = 8·(1/2)/5 = 4/5.
Question 6: What is the period of y = tan(πx)?
- 1 (Correct answer)
- 2
- π
- 1/π
Correct answer: 1
The period of tan(Bx) is π/|B|; here B = π, so period = π/π = 1.
Question 7: Which expression is equivalent to sin²(θ) using a power-reduction identity?
- (1 - cos(2θ))/2 (Correct answer)
- (1 + cos(2θ))/2
- (1 - cos(θ))/2
- (cos(2θ) - 1)/2
Correct answer: (1 - cos(2θ))/2
The power-reduction identity states sin²θ = (1 - cos(2θ))/2.
PRE CALCULUS Trigonometry 3
Using the sum formula, what is the exact value of sin(75°)?