To solve the equation [tex]\( \cos^2(x) = 2 + 2\sin(x) \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], we can proceed step-by-step as follows:
1. Rewrite the equation with trigonometric identities:
Recall that [tex]\( \cos^2(x) = 1 - \sin^2(x) \)[/tex]. Using this identity, we rewrite the equation:
[tex]\[
1 - \sin^2(x) = 2 + 2\sin(x)
\][/tex]
2. Form a quadratic equation:
Rearrange the equation to bring all terms to one side:
[tex]\[
1 - \sin^2(x) - 2\sin(x) - 2 = 0
\][/tex]
Simplify it:
[tex]\[
-\sin^2(x) - 2\sin(x) - 1 = 0
\][/tex]
Multiply by [tex]\(-1\)[/tex] to make it easier to solve:
[tex]\[
\sin^2(x) + 2\sin(x) + 1 = 0
\][/tex]
3. Solve the quadratic equation:
Let [tex]\( u = \sin(x) \)[/tex]. The quadratic equation becomes:
[tex]\[
u^2 + 2u + 1 = 0
\][/tex]
Factorize the quadratic equation:
[tex]\[
(u + 1)^2 = 0
\][/tex]
So, we have:
[tex]\[
u + 1 = 0
\][/tex]
Solving for [tex]\( u \)[/tex]:
[tex]\[
u = -1
\][/tex]
4. Find the corresponding angle for [tex]\( \sin(x) = -1 \)[/tex]:
We need to find [tex]\( x \)[/tex] such that [tex]\( \sin(x) = -1 \)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex]. The value [tex]\( \sin(x) = -1 \)[/tex] occurs at:
[tex]\[
x = \frac{3\pi}{2}
\][/tex]
Therefore, the solution to the equation [tex]\( \cos^2(x) = 2 + 2\sin(x) \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] is:
[tex]\[
x = \frac{3\pi}{2}
\][/tex]
