How to solve csc^2 x = 5 in the 0 <= x < 2pi?

To solve the equation csc^2 x = 5 in the range 0 <= x < 2pi, you can follow these steps:

Step 1: Rewrite the equation
Since csc^2 x is the reciprocal of sin^2 x, you can rewrite the equation as sin^2 x = 1/5.

Step 2: Solve for sin x
Take the square root of both sides to solve for sin x: sin x = ±√(1/5).

Step 3: Find the values of x
Remember that sin x is positive in the first and second quadrants, and negative in the third and fourth quadrants.

In the first quadrant (0 to π/2), sin x = √(1/5). Use the inverse sin function (sin^(-1)) to find the value of x: x = sin^(-1)(√(1/5)).

In the second quadrant (π/2 to π), sin x = √(1/5). Use the inverse sin function to find the value of x: x = π - sin^(-1)(√(1/5)).

In the third quadrant (π to 3π/2), sin x = -√(1/5). Use the inverse sin function to find the value of x: x = π + sin^(-1)(√(1/5)).

In the fourth quadrant (3π/2 to 2π), sin x = -√(1/5). Use the inverse sin function to find the value of x: x = 2π - sin^(-1)(√(1/5)).

Remember that the values of x must fall within the range 0 <= x < 2π. So check if any of the values obtained are outside of this range and modify them accordingly.

By following these steps, you can find the solution to the equation csc^2 x = 5 in the given range.