Let's solve the given problems step-by-step.
### Problem 1: Finding the Exact Value of [tex]\(\tan^{-1}\left[\tan \left(\frac{11 \pi}{6}\right)\right]\)[/tex]
1. Given Angle: [tex]\(\frac{11 \pi}{6}\)[/tex] radians.
2. Calculate [tex]\(\tan\)[/tex] of the Angle:
[tex]\[
\tan \left(\frac{11 \pi}{6}\right) \approx -0.5773502691896264
\][/tex]
3. Inverse Tangent ([tex]\(\tan^{-1}\)[/tex]):
[tex]\[
\tan^{-1}(-0.5773502691896264) \approx -0.5235987755982994 \text{ radians}
\][/tex]
Hence, the exact value of [tex]\(\tan^{-1}\left[\tan \left(\frac{11 \pi}{6}\right)\right]\)[/tex] is [tex]\(-0.5235987755982994\)[/tex] radians.
### Problem 2: Solve the Equation [tex]\(12 \sin x \cos x + 9 \sin x = 8 \cos x + 6\)[/tex]
1. Equation to Solve:
[tex]\[
12 \sin x \cos x + 9 \sin x = 8 \cos x + 6
\][/tex]
2. Finding Solutions in [tex]\([0, 2\pi)\)[/tex]:
The solutions to this equation are:
[tex]\[
x = 2 \text{atan} \left(\sqrt{7}\right), 2 \text{atan} \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right), -2 \text{atan} \left(\sqrt{7}\right), 2 \text{atan} \left(\frac{\sqrt{5}}{2} + \frac{3}{2}\right)
\][/tex]
3. Valid Solutions Within Interval [tex]\([0, 2\pi)\)[/tex]:
Filtering and rounding the values in the given interval:
[tex]\[
x \approx 2.4189, 0.7297, 2.4119 \text{ (rounded to 4 decimal places)}
\][/tex]
Thus, the solutions to the equation [tex]\(12 \sin x \cos x + 9 \sin x = 8 \cos x + 6\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] are approximately [tex]\(2.4189, 0.7297, 2.4119\)[/tex] (rounded to 4 decimal places).
