To determine the solutions to the equation [tex]\(8 \cos^2 \theta - 3 \cos \theta = 0\)[/tex] in the interval [tex]\(0^\circ \leq \theta \leq 180^\circ\)[/tex], follow these steps:
1. Rewrite the equation in a simpler form:
Let [tex]\( x = \cos \theta \)[/tex]. The equation becomes:
[tex]\[
8x^2 - 3x = 0
\][/tex]
2. Factor the quadratic equation:
[tex]\[
x(8x - 3) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 0 \quad \text{or} \quad 8x - 3 = 0
\][/tex]
Solving [tex]\( 8x - 3 = 0 \)[/tex]:
[tex]\[
8x = 3 \implies x = \frac{3}{8}
\][/tex]
4. Convert back to [tex]\( \theta \)[/tex]:
Recall [tex]\( x = \cos \theta \)[/tex].
- For [tex]\( \cos \theta = 0 \)[/tex]:
[tex]\[
\theta = 90^\circ
\][/tex]
- For [tex]\( \cos \theta = \frac{3}{8} \)[/tex]:
[tex]\[
\theta \approx 68.0^\circ
\][/tex]
Therefore, the possible values of [tex]\( \theta \)[/tex] in the interval [tex]\([0^\circ, 180^\circ]\)[/tex] are [tex]\( 90.0^\circ \)[/tex] and [tex]\( 68.0^\circ \)[/tex].
Checking the given choices:
- [tex]\( 0.0^\circ \)[/tex]: Not a solution.
- [tex]\( 22.0^\circ \)[/tex]: Not a solution.
- [tex]\( 52.0^\circ \)[/tex]: Not a solution.
- [tex]\( 68.0^\circ \)[/tex]: Solution.
- [tex]\( 90.0^\circ \)[/tex]: Solution.
Thus, the solutions are [tex]\( 68.0^\circ \)[/tex] and [tex]\( 90.0^\circ \)[/tex].
