To determine which values of [tex]\( x \)[/tex] are solutions to the given equation
[tex]\[
(6 \cos(x) - 5 \sin(x))^2 + 11 \sin^2(x) = 36,
\][/tex]
we go through the following steps:
1. Express the given equation:
The equation we need to solve is:
[tex]\[
(6 \cos(x) - 5 \sin(x))^2 + 11 \sin^2(x) = 36.
\][/tex]
2. Set up the equation:
Let's denote the left-hand side of the equation as:
[tex]\[
f(x) = (6 \cos(x) - 5 \sin(x))^2 + 11 \sin^2(x).
\][/tex]
3. Possible values of [tex]\( x \)[/tex]:
Based on solving the equation, we find that the solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = 0, \quad x = -\frac{\pi}{2}, \quad x = \frac{\pi}{2}.
\][/tex]
These values of [tex]\( x \)[/tex] make the equation true when substituted back. Hence, the values of [tex]\( x \)[/tex] that are solutions to the given trigonometric equation are:
[tex]\[
x = 0, \quad x = -\frac{\pi}{2}, \quad x = \frac{\pi}{2}.
\][/tex]
Thus, the correct options are:
[tex]\( x = 0, -\frac{\pi}{2}, \frac{\pi}{2} \)[/tex].
