To find the general solution to the trigonometric equation [tex]\(2 \cos^2 \theta - \cos \theta = 0\)[/tex], let's go through the steps in detail:
1. Factor the equation:
The given equation is:
[tex]\[
2 \cos^2 \theta - \cos \theta = 0
\][/tex]
Factor out [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta (2 \cos \theta - 1) = 0
\][/tex]
2. Set each factor to zero and solve:
For [tex]\(\cos \theta = 0\)[/tex]:
[tex]\[
\cos \theta = 0
\][/tex]
The angles [tex]\(\theta\)[/tex] where [tex]\(\cos \theta = 0\)[/tex] can be found using the unit circle. These angles are:
[tex]\[
\theta = 90^\circ + 360^\circ k \quad \text{or} \quad \theta = 270^\circ + 360^\circ k \quad \text{where } k \in \mathbb{Z}
\][/tex]
For [tex]\(2 \cos \theta - 1 = 0\)[/tex]:
[tex]\[
2 \cos \theta - 1 = 0 \quad \Rightarrow \quad 2 \cos \theta = 1 \quad \Rightarrow \quad \cos \theta = \frac{1}{2}
\][/tex]
The angles [tex]\(\theta\)[/tex] where [tex]\(\cos \theta = \frac{1}{2}\)[/tex] can be found using the unit circle. These angles are:
[tex]\[
\theta = 60^\circ + 360^\circ k \quad \text{or} \quad \theta = 300^\circ + 360^\circ k \quad \text{where } k \in \mathbb{Z}
\][/tex]
3. Combine all general solutions:
Therefore, the general solutions to the equation [tex]\(2 \cos^2 \theta - \cos \theta = 0\)[/tex] are:
[tex]\[
\theta = 90^\circ + 360^\circ k \quad \text{or} \quad \theta = 270^\circ + 360^\circ k \quad \text{or} \quad \theta = 60^\circ + 360^\circ k \quad \text{or} \quad \theta = 300^\circ + 360^\circ k
\][/tex]
where [tex]\(k \in \mathbb{Z}\)[/tex].
4. Match the solution with the choices provided:
Comparing with the given choices:
None of the choices exactly match the derived general solutions. Let's identify the proper captures:
a. C. [tex]$\theta=90^{\circ}+360 k$[/tex] or [tex]$\theta=270^{\circ}+360 k$[/tex] and [tex]$\theta=60^{\circ}+360 k$[/tex] or [tex]$\theta=300^{\circ}+360 k$[/tex] where [tex]$k \in Z$[/tex]
Does precisely match our combination of solutions.
Thus, the correct answer is:
a. [tex]\( \theta=90^\circ + 360k \text{ or } \theta=270^\circ + 360k \text{ and } \theta=60^\circ + 360k \text{ or } \theta=300^\circ + 360k \text{ where } k \in \mathbb{Z} \)[/tex]
