TMUA Trigonometry 4 — Questions and Answers
Question 1: In a triangle with sides a = 5, b = 7, and included angle C = 60°, what is side c?
- √39 (Correct answer)
- √74
- √61
- 6
Correct answer: √39
By the cosine rule: c² = 25 + 49 - 2(5)(7)cos60° = 74 - 35 = 39, so c = √39.
Question 2: Which of the following is the correct expression for the area of a triangle with sides a, b and included angle C?
- (1/2)ab·sin(C) (Correct answer)
- (1/2)ab·cos(C)
- ab·sin(C)
- (1/2)a²·sin(C)
Correct answer: (1/2)ab·sin(C)
The area formula using two sides and the included angle is Area = (1/2)ab·sin(C).
Question 3: What is the value of arcsin(sin(5π/6))?
- π/6 (Correct answer)
- 5π/6
- -π/6
- π
Correct answer: π/6
sin(5π/6) = 1/2, and arcsin returns values in [-π/2, π/2], so arcsin(1/2) = π/6.
Question 4: If tan(x) = -√3 and x ∈ [π/2, π], what is x?
- 2π/3 (Correct answer)
- 5π/3
- π/3
- 4π/3
Correct answer: 2π/3
tan(x) = -√3 and x is in Q2, so x = π - π/3 = 2π/3.
Question 5: Which identity correctly represents cos²(x) in terms of cos(2x)?
- (1 + cos(2x))/2 (Correct answer)
- (1 - cos(2x))/2
- cos(2x)/2
- 1 - cos²(2x)
Correct answer: (1 + cos(2x))/2
From cos(2x) = 2cos²x - 1, rearranging gives cos²x = (1 + cos2x)/2.
Question 6: Given that sin(A) = 5/13 and cos(B) = 4/5, with A and B both in the first quadrant, find sin(A + B).
- 56/65 (Correct answer)
- 63/65
- 33/65
- 16/65
Correct answer: 56/65
cos A = 12/13, sin B = 3/5; sin(A+B) = sinAcosB + cosAsinB = (5/13)(4/5) + (12/13)(3/5) = 20/65 + 36/65 = 56/65.
Question 7: What is the number of solutions to sin(2x) = 0.5 in the interval [0, 2π]?
- 4 (Correct answer)
- 2
- 1
- 6
Correct answer: 4
Let u = 2x ∈ [0, 4π]; sin(u) = 0.5 has 4 solutions: π/6, 5π/6, π/6+2π, 5π/6+2π.
TMUA Trigonometry 4
In a triangle with sides a = 5, b = 7, and included angle C = 60°, what is side c?