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 ) = \square \\)
(type an exact answer, using radicals as needed.)

Explanation:

Step1: Find \(\cos\theta\)

Using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), we have \(\cos\theta=\sqrt{1-\sin^{2}\theta}\).
Since \(\sin\theta=\frac{5}{6}\), then \(\cos\theta=\sqrt{1 - (\frac{5}{6})^{2}}=\sqrt{1-\frac{25}{36}}=\sqrt{\frac{11}{36}}=\frac{\sqrt{11}}{6}\).

Step2: Use the double - angle formula for \(\sin(2\theta)\)

The double - angle formula is \(\sin(2\theta)=2\sin\theta\cos\theta\).
Substitute \(\sin\theta=\frac{5}{6}\) and \(\cos\theta=\frac{\sqrt{11}}{6}\) into the formula:
\(\sin(2\theta)=2\times\frac{5}{6}\times\frac{\sqrt{11}}{6}\).

Step3: Simplify the expression

\(\sin(2\theta)=\frac{10\sqrt{11}}{36}=\frac{5\sqrt{11}}{18}\).

Answer:

\(\frac{5\sqrt{11}}{18}\)