Given [tex]\(\sin \theta = \frac{1}{2}\)[/tex] and [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex], we need to find the values of [tex]\(\cos \theta\)[/tex] and [tex]\(\tan \theta\)[/tex].
Let's analyze this in a step-by-step manner:
1. Identify the Quadrant:
- The range given for [tex]\(\theta\)[/tex] is [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex], which means [tex]\(\theta\)[/tex] is in the second quadrant.
2. Sine and Cosine Values in the Second Quadrant:
- In the second quadrant, [tex]\(\sin \theta\)[/tex] is positive, and [tex]\(\cos \theta\)[/tex] is negative.
3. Calculate [tex]\(\cos \theta\)[/tex]:
- We use the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
- Substituting [tex]\(\sin \theta = \frac{1}{2}\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^2 + \cos^2 \theta = 1 \implies \frac{1}{4} + \cos^2 \theta = 1 \implies \cos^2 \theta = 1 - \frac{1}{4} = \frac{3}{4}
\][/tex]
- Therefore:
[tex]\[
\cos \theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}
\][/tex]
- Since [tex]\(\theta\)[/tex] is in the second quadrant where [tex]\(\cos \theta\)[/tex] is negative:
[tex]\[
\cos \theta = -\frac{\sqrt{3}}{2}
\][/tex]
4. Calculate [tex]\(\tan \theta\)[/tex]:
- We use the definition of tangent:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
- Substituting the values we have:
[tex]\[
\tan \theta = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{2}{-\sqrt{3}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
\][/tex]
Thus, the values of [tex]\(\cos \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] are:
[tex]\[
\cos \theta = -\frac{\sqrt{3}}{2}, \quad \tan \theta = -\frac{\sqrt{3}}{3}
\][/tex]
So, the correct answer is:
[tex]\[
\cos \theta = -\frac{\sqrt{3}}{2} ; \tan \theta = -\frac{\sqrt{3}}{3}
\][/tex]
