Alright, let's tackle this problem step-by-step.
Given:
[tex]\[
\cos \left(\theta + 30^\circ\right) = \frac{1}{2} \sin \theta
\][/tex]
We will use the angle addition formula for the cosine function:
[tex]\[
\cos \left(\theta + 30^\circ\right) = \cos \theta \cos 30^\circ - \sin \theta \sin 30^\circ
\][/tex]
We know from trigonometric identities that:
[tex]\[
\cos 30^\circ = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin 30^\circ = \frac{1}{2}
\][/tex]
Substituting these values into the angle addition formula, we get:
[tex]\[
\cos \left(\theta + 30^\circ\right) = \cos \theta \cdot \frac{\sqrt{3}}{2} - \sin \theta \cdot \frac{1}{2}
\][/tex]
Let's substitute this back into the given equation:
[tex]\[
\cos \theta \cdot \frac{\sqrt{3}}{2} - \sin \theta \cdot \frac{1}{2} = \frac{1}{2} \sin \theta
\][/tex]
Now, we need to simplify this equation:
[tex]\[
\cos \theta \cdot \frac{\sqrt{3}}{2} - \sin \theta \cdot \frac{1}{2} - \frac{1}{2} \sin \theta = 0
\][/tex]
Combine the [tex]\(\sin \theta\)[/tex] terms:
[tex]\[
\cos \theta \cdot \frac{\sqrt{3}}{2} - \sin \theta \left(\frac{1}{2} + \frac{1}{2}\right) = 0
\][/tex]
This simplifies to:
[tex]\[
\cos \theta \cdot \frac{\sqrt{3}}{2} - \sin \theta \cdot 1 = 0
\][/tex]
or:
[tex]\[
\cos \theta \cdot \frac{\sqrt{3}}{2} = \sin \theta
\][/tex]
Rearranging the terms gives:
[tex]\[
\cos \theta \cdot \frac{\sqrt{3}}{2} = \sin \theta
\][/tex]
To isolate [tex]\(\tan \theta\)[/tex], divide both sides by [tex]\(\cos \theta\)[/tex]:
[tex]\[
\frac{\sqrt{3}}{2} = \frac{\sin \theta}{\cos \theta}
\][/tex]
Recall that [tex]\(\frac{\sin \theta}{\cos \theta} = \tan \theta\)[/tex]:
So, we have:
[tex]\[
\tan \theta = \frac{\sqrt{3}}{2}
\][/tex]
Hence, we have shown that:
[tex]\[
\tan \theta = \frac{\sqrt{3}}{2}
\][/tex]
This completes the solution.
