10. Solve each of the following for the given domain. Express each answer in the indicated
unit value.
a. csc 𝜃 = √2 , 0 ≤ 𝜃 ≤ 4𝜋
In order to solve for 𝜃, we need to find the angle at which the cosecant function is equal to √2 within the given domain.
We can start by finding the reference angle that has a cosecant value of √2. The reference angle θ' is such that csc(θ') = √2. Since cosecant is the reciprocal of sine, this means sin(θ') = 1/√2 = √2/2.
We know that the sine function is positive in the first and second quadrants, so we can determine that the reference angle is 45 degrees or π/4 radians.
Now, we need to find the angles in the given domain that have a cosecant of √2. Since the sine function is positive in the first and second quadrants, the angles in the domain where the cosecant function is equal to √2 are:
θ = π/4, 3π/4, 5π/4, 7π/4
So, the solutions within the given domain are:
𝜃 = π/4, 3π/4, 5π/4, 7π/4