PRE CALCULUS Trigonometry 4 — Questions and Answers
Question 1: What is the exact value of cos(7π/4)?
- √2/2 (Correct answer)
- -√2/2
- √3/2
- -√3/2
Correct answer: √2/2
7π/4 is in the fourth quadrant with reference angle π/4, and cosine is positive in Q4, so cos(7π/4) = √2/2.
Question 2: Which of the following is an odd function?
- f(x) = sin(x) (Correct answer)
- f(x) = cos(x)
- f(x) = sec(x)
- f(x) = cos²(x)
Correct answer: f(x) = sin(x)
sin(-x) = -sin(x) satisfies the definition of an odd function, f(-x) = -f(x).
Question 3: For the equation 2sin(x) - 1 = 0 on [0, 2π), the solutions are:
- π/6 and 5π/6 (Correct answer)
- π/3 and 2π/3
- π/6 and 7π/6
- π/4 and 3π/4
Correct answer: π/6 and 5π/6
2sin(x) = 1 gives sin(x) = 1/2, which is true at x = π/6 and x = π - π/6 = 5π/6 in [0, 2π).
Question 4: The product-to-sum formula converts sin(A)cos(B) into:
- [sin(A+B) + sin(A-B)]/2 (Correct answer)
- [cos(A-B) - cos(A+B)]/2
- [sin(A+B) - sin(A-B)]/2
- [cos(A+B) + cos(A-B)]/2
Correct answer: [sin(A+B) + sin(A-B)]/2
The product-to-sum identity gives sin(A)cos(B) = [sin(A+B) + sin(A-B)]/2.
Question 5: A point on the unit circle at angle θ has coordinates (cos θ, sin θ). At θ = 2π/3, the coordinates are:
- (-1/2, √3/2) (Correct answer)
- (1/2, √3/2)
- (-√3/2, 1/2)
- (-1/2, -√3/2)
Correct answer: (-1/2, √3/2)
2π/3 = 180° - 60°; in Q2, cosine is negative and sine is positive, giving (-1/2, √3/2).
Question 6: What is the domain of y = arccos(x)?
- [-1, 1] (Correct answer)
- [-π/2, π/2]
- (-∞, ∞)
- [0, π]
Correct answer: [-1, 1]
arccos(x) is only defined for inputs x in [-1, 1] because cosine only produces values in that range.
Question 7: Which identity can be used to verify that sec²(θ) - 1 = tan²(θ)?
- The Pythagorean identity 1 + tan²θ = sec²θ (Correct answer)
- The reciprocal identity secθ = 1/cosθ
- The double-angle identity cos(2θ) = 1 - 2sin²θ
- The co-function identity sinθ = cos(π/2 - θ)
Correct answer: The Pythagorean identity 1 + tan²θ = sec²θ
Rearranging 1 + tan²θ = sec²θ directly gives sec²θ - 1 = tan²θ.
PRE CALCULUS Trigonometry 4
What is the exact value of cos(7π/4)?