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 solve the equation 1 - sin(x) = 3cos^2(x) over the domain 0 degrees to 360 degrees, we can follow these steps:

Step 1: Simplify the equation using trigonometric identities.

Start with the equation: 1 - sin(x) = 3cos^2(x)

We know that cos^2(x) = 1 - sin^2(x) (using the identity cos^2(x) + sin^2(x) = 1)

Substituting this into the equation, we get: 1 - sin(x) = 3(1 - sin^2(x))

Expand: 1 - sin(x) = 3 - 3sin^2(x)

Rearrange the terms: 3sin^2(x) - sin(x) + 2 = 0

Step 2: Solve the quadratic equation.

Let's rewrite the equation as 3sin^2(x) - sin(x) + 2 = 0

Now we need to solve this quadratic equation for sin(x). We can factor or use the quadratic formula to find the solutions.

Factoring is not straightforward, so we'll use the quadratic formula:

sin(x) = (-b ± sqrt(b^2 - 4ac)) / (2a)

In our case, a = 3, b = -1, and c = 2.

sin(x) = (-(-1) ± sqrt((-1)^2 - 4(3)(2))) / (2(3))
= (1 ± sqrt(1 - 24)) / 6
= (1 ± sqrt(-23)) / 6

Step 3: Determine the reference angle.

Since sin(x) can only be between -1 and 1, the equation sin(x) = (1 ± sqrt(-23)) / 6 has no real solutions. Therefore, there are no solutions in the specified domain (0 degrees to 360 degrees) that satisfy the equation 1 - sin(x) = 3cos^2(x).