Question

if ( sin \theta=\frac{5}{6}, 0<\theta<\frac{pi}{2} ), 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{5 sqrt{11}}{18} ) (type an exact answer, using radicals as needed.) (b) ( cos (2 \theta)=-\frac{7}{18} ) (type an exact answer, using radicals as needed.) (c) ( sin \frac{\theta}{2}=\frac{sqrt{18-3 sqrt{11}}}{6} ) (type an exact answer, using radicals as needed.) (d) ( cos \frac{\theta}{2}=square ) (type an exact answer, using radicals as needed.)

Explanation:

Step1: Find $\cos\theta$

Since $\sin^{2}\theta+\cos^{2}\theta = 1$ and $\sin\theta=\frac{5}{6}$, then $\cos\theta=\sqrt{1-\sin^{2}\theta}=\sqrt{1 - (\frac{5}{6})^{2}}=\sqrt{\frac{36 - 25}{36}}=\frac{\sqrt{11}}{6}$.

Step2: Use the half - angle formula for $\cos\frac{\theta}{2}$

The half - angle formula is $\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}$. Here $\alpha=\theta$, so $\cos\frac{\theta}{2}=\sqrt{\frac{1+\cos\theta}{2}}$.
Substitute $\cos\theta=\frac{\sqrt{11}}{6}$ into the formula:

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Answer:

$\frac{\sqrt{18 + 3\sqrt{11}}}{6}$