TMUA Trigonometry 5 — Questions and Answers
Question 1: What is the value of cos(A - B) if sin(A) = 3/5, cos(B) = 5/13, and both A and B are acute?
- 56/65 (Correct answer)
- 33/65
- 63/65
- 16/65
Correct answer: 56/65
cos A = 4/5, sin B = 12/13; cos(A-B) = cosAcosB + sinAsinB = (4/5)(5/13)+(3/5)(12/13) = 20/65+36/65 = 56/65.
Question 2: Which of the following is a solution to cos(2x) + cos(x) = 0 in [0, 2π]?
- x = π/3 (Correct answer)
- x = π/6
- x = π/4
- x = π/8
Correct answer: x = π/3
Substituting 2cos²x - 1 + cosx = 0 gives (2cosx - 1)(cosx + 1) = 0, so cosx = 1/2 → x = π/3.
Question 3: The graph of y = sin(x) is translated π/4 to the right. What is the new equation?
- y = sin(x - π/4) (Correct answer)
- y = sin(x + π/4)
- y = sin(x) - π/4
- y = sin(x - π/2)
Correct answer: y = sin(x - π/4)
A rightward shift by c replaces x with x - c, giving y = sin(x - π/4).
Question 4: What is the exact value of sin(3π/4)?
- √2/2 (Correct answer)
- -√2/2
- √3/2
- 1/2
Correct answer: √2/2
sin(3π/4) = sin(π - π/4) = sin(π/4) = √2/2.
Question 5: Which of the following identities is INCORRECT?
- sin²x + cos²x = 2 (Correct answer)
- tan²x + 1 = sec²x
- 1 + cot²x = csc²x
- cos(2x) = cos²x - sin²x
Correct answer: sin²x + cos²x = 2
The Pythagorean identity is sin²x + cos²x = 1, not 2; the other three identities are correct.
Question 6: A right triangle has hypotenuse 10 and one angle of 30°. What is the length of the side opposite the 30° angle?
- 5 (Correct answer)
- 5√3
- 10√3
- 5√2
Correct answer: 5
The side opposite 30° = hypotenuse × sin(30°) = 10 × 0.5 = 5.
Question 7: For f(x) = 2cos(x/2), what is the period?
- 4π (Correct answer)
- π
- 2π
- π/2
Correct answer: 4π
The period of cos(kx) is 2π/k; with k = 1/2 the period is 2π/(1/2) = 4π.
TMUA Trigonometry 5
What is the value of cos(A - B) if sin(A) = 3/5, cos(B) = 5/13, and both A and B are acute?