find, to the nearest minute, the solution set of 1- sin x = 3 cos^2 x over the domain 0 degrees to 360 degres

To find the solution set of the equation 1 - sin x = 3 cos^2 x over the domain of 0 degrees to 360 degrees, we can follow these steps:

Step 1: Simplify the equation:
1 - sin x = 3 cos^2 x

Step 2: Rewrite cos^2 x in terms of sin x using the identity: cos^2 x = 1 - sin^2 x
1 - sin x = 3(1 - sin^2 x)

Step 3: Expand and rearrange the equation:
1 - sin x = 3 - 3sin^2 x
3sin^2 x - sin x - 2 = 0

Step 4: Solve the quadratic equation:
Factor the equation: (3sin x + 2)(sin x - 1) = 0

Setting each factor to zero and solving:
3sin x + 2 = 0
sin x = -2/3

sin x - 1 = 0
sin x = 1

Step 5: Solve for the values of x within the given domain:
0 degrees to 360 degrees.

For sin x = -2/3:
To find the value of x in the given domain, we can use the inverse sine function (sin^{-1}).

Using a calculator or a table of values, sin^{-1}(-2/3) is approximately -41.81 degrees and -138.19 degrees.

For sin x = 1:
Using the inverse sine function, sin^{-1}(1) is equal to 90 degrees.

Therefore, the solution set within the given domain is:
x = -41.81 degrees, -138.19 degrees, and 90 degrees (rounded to the nearest minute).