Q. If \( y = \sin^{-1}(x) + \cos^{-1}(x) \), what is the value of \( y \)?
A.
0
B.
1
C.
\( \frac{\pi}{2} \)
D.
undefined
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Solution
Since \( \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \) for all \( x \) in the domain of \( \sin^{-1} \) and \( \cos^{-1} \), the answer is \( \frac{\pi}{2} \).
Correct Answer: C — \( \frac{\pi}{2} \)
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Q. If \( y = \tan^{-1}(x) + \tan^{-1}(y) \), what is the value of \( y \) when \( x = 1 \)?
A.
0
B.
1
C.
\( \frac{\pi}{4} \)
D.
undefined
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Solution
When \( x = 1 \), \( y = \tan^{-1}(1) + \tan^{-1}(y) \) leads to \( y = \frac{\pi}{4} \).
Correct Answer: C — \( \frac{\pi}{4} \)
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Q. The range of sin^(-1)(x) is:
A.
[-π/2, π/2]
B.
[0, π]
C.
[-1, 1]
D.
[0, 1]
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Solution
The range of sin^(-1)(x) is [-π/2, π/2].
Correct Answer: A — [-π/2, π/2]
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Q. The range of the function y = sin^(-1)(x) is:
A.
(0, π)
B.
[-π/2, π/2]
C.
[-1, 1]
D.
[0, 1]
Show solution
Solution
The range of y = sin^(-1)(x) is [-π/2, π/2].
Correct Answer: B — [-π/2, π/2]
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Q. The value of cos(tan^(-1)(x)) is:
A.
1/√(1+x^2)
B.
x/√(1+x^2)
C.
√(1+x^2)/x
D.
0
Show solution
Solution
Using the right triangle definition, cos(tan^(-1)(x)) = adjacent/hypotenuse = 1/√(1+x^2).
Correct Answer: A — 1/√(1+x^2)
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Q. The value of sin(tan^(-1)(x)) is:
A.
x/√(1+x^2)
B.
√(1-x^2)
C.
1/√(1+x^2)
D.
x
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Solution
Using the right triangle definition, sin(tan^(-1)(x)) = opposite/hypotenuse = x/√(1+x^2).
Correct Answer: A — x/√(1+x^2)
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Q. The value of sin^(-1)(sin(π/4)) is:
A.
π/4
B.
3π/4
C.
0
D.
π/2
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Solution
Since π/4 is in the range of sin^(-1), sin^(-1)(sin(π/4)) = π/4.
Correct Answer: A — π/4
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Q. What is the derivative of \( y = \tan^{-1}(x) \)?
A.
\( \frac{1}{1+x^2} \)
B.
\( \frac{1}{x^2+1} \)
C.
\( \frac{1}{x} \)
D.
0
Show solution
Solution
The derivative of \( y = \tan^{-1}(x) \) is \( \frac{1}{1+x^2} \).
Correct Answer: A — \( \frac{1}{1+x^2} \)
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Q. What is the principal value of cot^(-1)(0)?
A.
0
B.
π/2
C.
π
D.
undefined
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Solution
cot^(-1)(0) = π/2
Correct Answer: B — π/2
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Q. What is the principal value of sec^(-1)(2)?
A.
π/3
B.
π/4
C.
π/6
D.
0
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Solution
sec^(-1)(2) = π/3, since sec(π/3) = 2.
Correct Answer: A — π/3
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Q. What is the range of the function sin^(-1)(x)?
A.
(-π/2, π/2)
B.
(-1, 1)
C.
(0, π)
D.
(0, 1)
Show solution
Solution
The range of sin^(-1)(x) is (-π/2, π/2).
Correct Answer: A — (-π/2, π/2)
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Q. What is the value of cos^(-1)(0)?
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Solution
cos^(-1)(0) corresponds to the angle where the cosine value is 0, which is π.
Correct Answer: C — π
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Q. What is the value of cot(cos^(-1)(1/2))?
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Solution
cot(cos^(-1)(1/2)) = √3
Correct Answer: A — √3
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Q. What is the value of sec^(-1)(2)?
A.
π/3
B.
π/4
C.
π/6
D.
0
Show solution
Solution
sec^(-1)(2) = π/3, since sec(π/3) = 2.
Correct Answer: A — π/3
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Q. What is the value of sin(tan^(-1)(1))?
A.
1/√2
B.
1/2
C.
1
D.
√2/2
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Solution
sin(tan^(-1)(1)) = √2/2
Correct Answer: D — √2/2
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Q. What is the value of sin(tan^(-1)(√3))?
A.
√3/2
B.
1/2
C.
1
D.
√2/2
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Solution
sin(tan^(-1)(√3)) = √3/2
Correct Answer: A — √3/2
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Q. What is the value of sin^(-1)(sin(π/4))?
A.
π/4
B.
3π/4
C.
π/2
D.
0
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Solution
sin^(-1)(sin(π/4)) = π/4, as π/4 is in the range of sin^(-1).
Correct Answer: A — π/4
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Q. What is the value of tan^(-1)(√3)?
A.
π/3
B.
π/4
C.
π/6
D.
π/2
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Solution
tan^(-1)(√3) corresponds to the angle π/3.
Correct Answer: A — π/3
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Q. What is the value of \( \tan(\tan^{-1}(3)) \)?
A.
0
B.
1
C.
3
D.
undefined
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Solution
By definition, \( \tan(\tan^{-1}(x)) = x \). Therefore, \( \tan(\tan^{-1}(3)) = 3 \).
Correct Answer: C — 3
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