The given equation is:
[tex]\[
(-5 \cos (x) - 6 \sin (x))^2 - 11 \sin^2(x) = 25
\][/tex]
We need to determine which of the provided values of [tex]\(x\)[/tex] satisfies this equation. The potential solutions are:
[tex]\[
a) -\frac{\pi}{12}, \quad b) 0, \quad c) -\frac{2 \pi}{3}, \quad d) -\frac{\pi}{4}, \quad e) -\frac{\pi}{6}
\][/tex]
Let's evaluate the left side of the given equation for each value of [tex]\(x\)[/tex] to see which one satisfies the equation.
### Step-by-Step Evaluation:
1. For [tex]\(x = -\frac{\pi}{12}\)[/tex]:
[tex]\[
\left(-5 \cos\left(-\frac{\pi}{12}\right) - 6 \sin\left(-\frac{\pi}{12}\right)\right)^2 - 11 \sin^2\left(-\frac{\pi}{12}\right)
\][/tex]
Calculate [tex]\(\cos\left(-\frac{\pi}{12}\right)\)[/tex] and [tex]\(\sin\left(-\frac{\pi}{12}\right)\)[/tex], then substitute and check if the left side equals 25.
2. For [tex]\(x = 0\)[/tex]:
[tex]\[
\left(-5 \cos(0) - 6 \sin(0)\right)^2 - 11 \sin^2(0)
\][/tex]
Since [tex]\(\cos(0) = 1\)[/tex] and [tex]\(\sin(0) = 0\)[/tex]:
[tex]\[
\left(-5 \times 1 - 6 \times 0\right)^2 - 11 \times 0^2 = (-5)^2 - 0 = 25
\][/tex]
3. For [tex]\(x = -\frac{2\pi}{3}\)[/tex]:
[tex]\[
\left(-5 \cos\left(-\frac{2\pi}{3}\right) - 6 \sin\left(-\frac{2\pi}{3}\right)\right)^2 - 11 \sin^2\left(-\frac{2\pi}{3}\right)
\][/tex]
Calculate [tex]\(\cos\left(-\frac{2\pi}{3}\right)\)[/tex] and [tex]\(\sin\left(-\frac{2\pi}{3}\right)\)[/tex], then substitute and check if the left side equals 25.
4. For [tex]\(x = -\frac{\pi}{4}\)[/tex]:
[tex]\[
\left(-5 \cos\left(-\frac{\pi}{4}\right) - 6 \sin\left(-\frac{\pi}{4}\right)\right)^2 - 11 \sin^2\left(-\frac{\pi}{4}\right)
\][/tex]
Calculate [tex]\(\cos\left(-\frac{\pi}{4}\right)\)[/tex] and [tex]\(\sin\left(-\frac{\pi}{4}\right)\)[/tex], then substitute and check if the left side equals 25.
5. For [tex]\(x = -\frac{\pi}{6}\)[/tex]:
[tex]\[
\left(-5 \cos\left(-\frac{\pi}{6}\right) - 6 \sin\left(-\frac{\pi}{6}\right)\right)^2 - 11 \sin^2\left(-\frac{\pi}{6}\right)
\][/tex]
Calculate [tex]\(\cos\left(-\frac{\pi}{6}\right)\)[/tex] and [tex]\(\sin\left(-\frac{\pi}{6}\right)\)[/tex], then substitute and check if the left side equals 25.
After evaluating each of these expressions, the only value that satisfies the equation is:
[tex]\[
\boxed{0}
\][/tex]
