How would I simplify and solve for X in this trigonometric equation? (Radians) cos(x+\pi )^(2) = sinx

Question

How would I simplify and solve for X in this trigonometric equation? (Radians)


`cos(x+\pi )^(2) = sinx`

Answer 1

Please, refer to the images below

Explanation:

We need to solve for x in the equation

cos (x+ pi) ^2 = sin (x)

cos (x+ pi) = - cos(x)

(-cos (x)) * (-cos (x)) = sin(x)

cos(x) ^2 = sin(x)

We know that

cos(x) ^2 + sin(x) ^2 = 1

cos(x) ^2 = 1 - sin(x) ^2

1 - sin(x) ^2 = sin(x)

sin(x) ^2 + sin (x) -1 = 0

Let A = sin(x)

A^2 + A - 1 = 0

(solutions attached in picture 1)

This means that

x = arcsin(A)

(solutions attached in picture 2)