Use the given information to find the exact value of the trigonometric function.
-$\sin \theta = - \frac { 3 } { 5 } , \quad \theta$ lies in quadrant IV Find $\sin \frac { \theta } { 2 }$.
A) $\frac { \sqrt { 10 } } { 10 }$
B) $\frac { \sqrt { 5 } } { 5 }$
C) $- \frac { \sqrt { 5 } } { 5 }$
D) $- \frac { \sqrt { 30 } } { 10 }$

Question

Quizplus · Accepted Answer

In quadrant IV, $\sin \theta < 0$ and $\cos \theta > 0$. The half-angle formula for sine is $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$. Since $\sin^2 \theta + \cos^2 \theta = 1$, we find $\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(-\frac{3}{5}\right)^2} = \sqrt{\frac{16}{25}} = \frac{4}{5}$. Because $\theta$ is in the fourth quadrant, $\cos \theta > 0$, so $\cos \theta = \frac{4}{5}$. Thus, $\sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{4}{5}}{2}} = \sqrt{\frac{1}{10}} = \frac{\sqrt{10}}{10}$.

Use the Given Information to Find the Exact Value of the Trigonometric