Problem 1 :
Domain of the function sin^{-1} x is ....
Solution:
Consider the given function as,
y = sin^{-1} x
The graph of sin^{-1} x or arc sin x is given as
-π/2 ≤ sin^{-1}x ≤ π/2
-1 ≤ sin^{-1}x ≤ 1
Hence, the domain of the function is [-1, 1].
Problem 2 :
Range of the function cos^{-1}x is ....
Solution:
Range of the function cos^{-1}x is [0, π].
Problem 3 :
The principal value of tan^{-1}√3 is .....
Solution:
Let y = tan^{-1}(√3)
tan y = √3
We know that the range of the principal value branch of tan^{-1}(-π/2, π/2).
tan y = tan(π/3)
y = π/3
Hence, the principal value of tan^{-1}(√3) is π/3.
Problem 4 :
Solution:
Problem 5 :
Principal values of the function tan^{-1}x lie in the interval....
Solution:
We know that range of principal value of tan^{-1}x is (-π/2, π/2).
Problem 6 :
Solution:
Problem 7 :
Solution:
Problem 8 :
Solution:
Problem 9 :
Solution:
Domain of sec^{-1}(1/2) is R - (-1, 1).
(i.e) (-∞, -1] ∪ [1, ∞)
So, no set of values exist for sec^{-1}(1/2).
Therefore, ∅ is the answer.
Problem 10 :
For x ∈ R, tan^{-1}(x^{2} + 1) + cot^{-1}(x^{2} + 1) is equal to ....
Solution:
Problem 11 :
If cos^{-1}(-x) = α - cos^{-1}x, then the value of α is ....
Solution:
We know that cos^{-1}(-x) = π - cos^{-1}x
By comparing, we get α = π
Hence, α = π.
Problem 12 :
Solution:
Given, |x| ≥ √2
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