(a) Evaluate cosec^-1(√2) (b) Evaluate tan^-1(sin(-π/2))

Question

There are two problems.

First one is Evaluate cosec^-1(√2)

And second one is Evaluate tan^-1(sin(-π/2))

Kindly solve this question.

Answer 1

(a) csc^-1(√2) is simply the angle θ at which csc(θ) = √2. Knowing that csc(θ) = 1/sin(θ), we are looking for the angle θ at which sin(θ) = 1/√2 (convince yourself of this). That angle is θ = π/4 (in quadrant 1). Thus,

csc^-1(√2) = π/4

(b) First we will evaluate sin(-π/2) which is simply equal to -1. Then, the problem becomes tan^-1(-1) which is the angle θ at which tan(θ) = -1. Using the common identity tan(θ) = sin(θ)/cos(θ), we are looking for the angle θ such that sin(θ)/cos(θ) = -1, or sin(θ) = -cos(θ). That angle is θ = 3π/4 (in quadrant 2). Thus,

tan^-1(sin(-π/2)) = 3π/4