Express the function as a trigonometric function of x.
-Find $\sin \theta$ if $\cos \theta = - \frac { \sqrt { 11 } } { 6 }$ and $\theta$ is in quadrant III.
A) $- \frac { \sqrt { 3 } } { 12 }$
B) $\frac { 5 } { 6 }$
C) $- \frac { 25 } { 36 }$
D) $- \frac { 5 } { 6 }$

Question

Quizplus · Accepted Answer

In quadrant III, both sine and cosine are negative. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we can find $\sin \theta$. Given $\cos \theta = - \frac{\sqrt{11}}{6}$, we have $\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(- \frac{\sqrt{11}}{6}\right)^2 = 1 - \frac{11}{36} = \frac{25}{36}$. Since $\theta$ is in quadrant III, $\sin \theta$ is negative, so $\sin \theta = - \frac{5}{6}$.

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