Trigonometry Inverse Functions 2 — Questions and Answers
Question 1: What is the range of arcsin(x)?
- [0, π]
- [-π/2, π/2] (Correct answer)
- (-π/2, π/2)
- (-∞, ∞)
Correct answer: [-π/2, π/2]
arcsin(x) outputs values in [-π/2, π/2] to make it a proper function.
Question 2: Evaluate arctan(√3).
- π/6
- π/3 (Correct answer)
- π/4
- 2π/3
Correct answer: π/3
tan(π/3) = √3, so arctan(√3) = π/3.
Question 3: Which of the following is NOT in the domain of arccos(x)?
- -1
- 0
- 1
- 2 (Correct answer)
Correct answer: 2
arccos(x) is only defined for x in [-1, 1], so x = 2 is outside the domain.
Question 4: Find the exact value of sin(arccos(3/5)).
- 3/5
- 4/5 (Correct answer)
- 5/3
- 5/4
Correct answer: 4/5
Using a 3-4-5 right triangle, if cos θ = 3/5 then sin θ = 4/5.
Question 5: What is the range of arccot(x)?
- (-π/2, π/2)
- (0, π) (Correct answer)
- [-π/2, π/2]
- [0, π]
Correct answer: (0, π)
arccot(x) is conventionally defined with range (0, π), excluding the endpoints.
Question 6: Simplify cos(arctan(x)) for x > 0.
- x/√(1+x²)
- 1/√(1+x²) (Correct answer)
- √(1+x²)
- x²/√(1+x²)
Correct answer: 1/√(1+x²)
If tan θ = x, then the hypotenuse is √(1+x²), making cos θ = 1/√(1+x²).
Question 7: Which identity holds for arcsin(x) + arccos(x)?
- 0
- π/4
- π/2 (Correct answer)
- π
Correct answer: π/2
arcsin(x) + arccos(x) = π/2 for all x in [-1, 1].
Trigonometry Inverse Functions 2
What is the range of arcsin(x)?