To solve the equation [tex]\(-8 \cos^2 \theta + 7 = 1\)[/tex] for [tex]\(\theta\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex], we follow these steps:
1. Simplify the equation:
Starting with the given equation:
[tex]\[
-8 \cos^2 \theta + 7 = 1
\][/tex]
Subtract 7 from both sides of the equation:
[tex]\[
-8 \cos^2 \theta = -6
\][/tex]
Divide both sides by -8:
[tex]\[
\cos^2 \theta = \frac{6}{8}
\][/tex]
Simplify the fraction:
[tex]\[
\cos^2 \theta = \frac{3}{4}
\][/tex]
2. Find the values of [tex]\(\cos \theta\)[/tex]:
Take the square root of both sides:
[tex]\[
\cos \theta = \pm \sqrt{\frac{3}{4}}
\][/tex]
Simplify the square root:
[tex]\[
\cos \theta = \pm \frac{\sqrt{3}}{2}
\][/tex]
3. Determine the angles [tex]\(\theta\)[/tex]:
The cosine function provides us with two principal values for each solution:
- When [tex]\(\cos \theta = \frac{\sqrt{3}}{2}\)[/tex]:
[tex]\[
\theta = \frac{\pi}{6} \quad \text{and} \quad \theta = \frac{11\pi}{6}
\][/tex]
- When [tex]\(\cos \theta = -\frac{\sqrt{3}}{2}\)[/tex]:
[tex]\[
\theta = \frac{5\pi}{6} \quad \text{and} \quad \theta = \frac{7\pi}{6}
\][/tex]
4. Convert these values into decimal form to match the provided results:
[tex]\[
\begin{aligned}
&\theta = \frac{\pi}{6} \approx 0.524 \\
&\theta = \frac{5\pi}{6} \approx 2.618 \\
&\theta = \frac{7\pi}{6} \approx 3.665 \\
&\theta = \frac{11\pi}{6} \approx 5.760 \\
\end{aligned}
\][/tex]
5. Arrange these solutions from least to greatest:
So the values of [tex]\(\theta\)[/tex] are:
[tex]\[
\theta = 0.523598775598299, 2.61799387799149, 3.66519142918809, \text{and } 5.75958653158129
\][/tex]
Thus, the solution in the correct order is:
[tex]\[
\theta = 0.523598775598299, 2.61799387799149, 3.66519142918809, 5.75958653158129
\][/tex]
