DOMAIN AND RANGE OF INVERSE TRIG FUNCTIONS

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^-1x ≤ π/2 

-1 ≤ sin^-1x ≤ 1

Hence, the domain of the function is [-1, 1].

Problem 2 :

Range of the function cos^-1x is ....

Solution:

Range of the function cos^-1x 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^-1x lie in the interval....

Solution:

We know that range of principal value of tan^-1x 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(x² + 1) + cot^-1(x² + 1) is equal to ....

Solution:

Problem 11 :

If cos^-1(-x) = α - cos^-1x, then the value of α is ....

Solution:

We know that cos^-1(-x) = π - cos^-1x

By comparing, we get α = π

Hence, α = π.

Problem 12 :

Solution:

Given, |x| ≥ √2