Let's solve the given problem through a step-by-step approach, addressing each part individually.
### Part A: Determining the exact value of [tex]\(\cos 2\theta\)[/tex]
Given:
[tex]\[
\sin \theta = \frac{3 \sqrt{2}}{5}
\][/tex]
and
[tex]\(
\frac{\pi}{2} < \theta < \pi
\)[/tex].
We need to find [tex]\(\cos 2\theta\)[/tex]. To do this, we will use the double angle identity for cosine:
[tex]\[
\cos 2\theta = 1 - 2\sin^2 \theta.
\][/tex]
First, compute [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\sin^2 \theta = \left(\frac{3 \sqrt{2}}{5}\right)^2 = \frac{(3 \sqrt{2})^2}{5^2} = \frac{18}{25}.
\][/tex]
Next, substitute [tex]\(\sin^2 \theta\)[/tex] into the double angle formula:
[tex]\[
\cos 2\theta = 1 - 2 \times \frac{18}{25} = 1 - \frac{36}{25} = \frac{25}{25} - \frac{36}{25} = \frac{25 - 36}{25} = \frac{-11}{25}.
\][/tex]
So, the exact value of [tex]\(\cos 2\theta\)[/tex] is:
[tex]\[
\cos 2\theta = -0.44.
\][/tex]
### Part B: Determining the exact value of [tex]\(\sin \left(\frac{\theta}{2}\right)\)[/tex]
To find [tex]\(\sin \left(\frac{\theta}{2}\right)\)[/tex], we use the half-angle identity for sine:
[tex]\[
\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}.
\][/tex]
We need to determine the correct sign for [tex]\(\sin \left(\frac{\theta}{2}\right)\)[/tex]. Since [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex], it means [tex]\(\frac{\pi}{4} < \frac{\theta}{2} < \frac{\pi}{2}\)[/tex]. In this range, [tex]\(\sin \left(\frac{\theta}{2}\right)\)[/tex] is positive.
First, we need to find [tex]\(\cos \theta\)[/tex]. We know:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1.
\][/tex]
Substitute [tex]\(\sin \theta = \frac{3 \sqrt{2}}{5}\)[/tex]:
[tex]\[
\left(\frac{3 \sqrt{2}}{5}\right)^2 + \cos^2 \theta = 1 \implies \frac{18}{25} + \cos^2 \theta = 1 \implies \cos^2 \theta = 1 - \frac{18}{25} = \frac{7}{25}.
\][/tex]
Since [tex]\(\cos \theta\)[/tex] in the interval [tex]\((\frac{\pi}{2}, \pi)\)[/tex] is negative:
[tex]\[
\cos \theta = -\sqrt{\frac{7}{25}} = -\frac{\sqrt{7}}{5}.
\][/tex]
Now use the half-angle identity:
[tex]\[
\sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} = \sqrt{\frac{1 - \left(-\frac{\sqrt{7}}{5}\right)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{7}}{5}}{2}} = \sqrt{\frac{\frac{5 + \sqrt{7}}{5}}{2}} = \sqrt{\frac{5 + \sqrt{7}}{10}}.
\][/tex]
So, the exact value of [tex]\(\sin \left(\frac{\theta}{2}\right)\)[/tex] is approximately:
[tex]\[
\sin \left(\frac{\theta}{2}\right) \approx 0.4852.
\][/tex]
Thus, the solutions are:
- [tex]\(\cos 2\theta = -0.44\)[/tex]
- [tex]\(\sin \left(\frac{\theta}{2}\right) \approx 0.4852\)[/tex].
