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Ch-2: Inverse Trigonometric Functions
12 Questions
Before You Practice Try each question on your own first. Use the Show Answer button only after you have selected your option, then read the explanation and revise the related concept from your chapter notes.
1. The principal value branch of sin⁻¹x is:
A. [-π/2, π/2]
B. [0, π]
C. (-π/2, π/2)
D. [0, 2π]
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Answer: A. [-π/2, π/2]
Explanation: By convention, the principal value branch of sin⁻¹x is restricted to [-π/2, π/2] to make the function single-valued.
2. The principal value branch of cos⁻¹x is:
A. [0, π]
B. [-π/2, π/2]
C. (0, π)
D. [-π, π]
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Answer: A. [0, π]
Explanation: The principal value branch of cos⁻¹x is defined as [0, π].
3. The domain of sin⁻¹x is:
A. [-1, 1]
B. The set of all real numbers
C. [0, 1]
D. (-∞, ∞)
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Answer: A. [-1, 1]
Explanation: Since sine values only range between -1 and 1, sin⁻¹x is defined only for x in [-1, 1].
4. The principal value branch of tan⁻¹x is:
A. (-π/2, π/2)
B. [0, π]
C. [-π/2, π/2]
D. (0, π)
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Answer: A. (-π/2, π/2)
Explanation: tan⁻¹x is defined for all real x with its principal value branch as the open interval (-π/2, π/2).
5. The value of sin⁻¹(sin(3π/4)) is:
A. π/4
B. 3π/4
C. -π/4
D. π/2
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Answer: A. π/4
Explanation: Since 3π/4 lies outside [-π/2, π/2], we use sin(3π/4) = sin(π - 3π/4) = sin(π/4), giving sin⁻¹(sin(3π/4)) = π/4.
6. tan⁻¹x + cot⁻¹x is equal to:
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Answer: A. π/2
Explanation: For all real x, tan⁻¹x and cot⁻¹x are complementary and always sum to π/2.
7. sin⁻¹x + cos⁻¹x is equal to:
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Answer: A. π/2
Explanation: For all x in [-1, 1], sin⁻¹x and cos⁻¹x are complementary angles summing to π/2.
8. The value of cos⁻¹(cos(2π/3)) is:
A. 2π/3
B. π/3
C. -2π/3
D. π
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Answer: A. 2π/3
Explanation: Since 2π/3 already lies within the principal value branch [0, π] of cos⁻¹x, the value remains 2π/3.
9. The range of cosec⁻¹x is:
A. [-π/2, π/2] excluding 0
B. [0, π]
C. (0, π)
D. [-π, π]
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Answer: A. [-π/2, π/2] excluding 0
Explanation: The principal value branch of cosec⁻¹x is [-π/2, π/2] with 0 excluded, since cosec x is undefined at x = 0.
10. sec⁻¹x is defined for values of x such that:
A. |x| ≥ 1
B. |x| ≤ 1
C. All real x
D. x > 0 only
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Answer: A. |x| ≥ 1
Explanation: Since sec x has range (-∞, -1] ∪ [1, ∞), sec⁻¹x is defined only when |x| ≥ 1.
11. The value of tan⁻¹(1) is:
A. π/4
B. π/2
C. π/3
D. π/6
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Answer: A. π/4
Explanation: Since tan(π/4) = 1 and π/4 lies within the principal branch, tan⁻¹(1) = π/4.
12. The identity 2 tan⁻¹x = tan⁻¹(2x/(1 - x²)) holds valid for:
A. |x| < 1
B. |x| > 1
C. All real x
D. x = 0 only
Show Answer
Answer: A. |x| < 1
Explanation: This double angle formula for tan⁻¹x is valid only within the restricted domain |x| < 1 to keep both sides consistent.
