Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$, given that $\cos \theta = \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.
A) $\frac { 5 } { 4 }$
B) $\frac { 3 \sqrt { 7 } } { 7 }$
C) $- \frac { 3 } { 2 }$
D) $- \frac { \sqrt { 5 } } { 3 }$

Question

Quizplus · Accepted Answer

In quadrant IV, sine is negative, and using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we can find $\sin \theta$. Given $\cos \theta = \frac{2}{3}$, we have $\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9}$. Taking the square root gives $\sin \theta = \pm \frac{\sqrt{5}}{3}$, but since $\theta$ is in quadrant IV where sine is negative, $\sin \theta = - \frac{\sqrt{5}}{3}$.

Use the Fundamental Identities to Find the Value of the Trigonometric