Question

given that $\cos\theta=-\frac{\sqrt{7}}{3},\frac{\pi}{2}<\theta<\pi$, find the exact value of each of the following. (a) $\sin(2\theta)$ (b) $\cos(2\theta)$ (c) $\sin\frac{\theta}{2}$ (d) $\cos\frac{\theta}{2}$ (a) $\sin(2\theta)=-\frac{2\sqrt{14}}{9}$ (simplify your answer, including any radicals. use integers or fractions for any numbers in the (b) $\cos(2\theta)=\frac{5}{9}$ (simplify your answer, including any radicals. use integers or fractions for any numbers in the (c) $\sin\frac{\theta}{2}=\sqrt{\frac{3 + \sqrt{7}}{6}}$ (simplify your answer, including any radicals. use integers or fractions for any numbers in the (d) $\cos\frac{\theta}{2}=\square$ (simplify your answer, including any radicals. use integers or fractions for any numbers in the

Explanation:

Step1: Determine the range of \(\frac{\theta}{2}\)

Since \(\frac{\pi}{2}<\theta <\pi\), then \(\frac{\pi}{4}<\frac{\theta}{2}<\frac{\pi}{2}\). In this range, \(\cos\frac{\theta}{2}>0\).

Step2: Use the half - angle formula for cosine

The half - angle formula for cosine is \(\cos\frac{\alpha}{2}=\sqrt{\frac{1 + \cos\alpha}{2}}\). Here \(\alpha=\theta\) and \(\cos\theta=-\frac{\sqrt{7}}{3}\).
Substitute \(\cos\theta\) into the formula:

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Answer:

\(\sqrt{\frac{3 - \sqrt{7}}{6}}\)