Complete the sentence so the result is an identity. Let x be any real number.
-Find $\cos \theta$ if $\csc \theta = - \frac { 5 } { 3 }$ and $\theta$ is in quadrant IV.
A) $\frac { 4 } { 5 }$
B) $\frac { 5 \sqrt { 29 } } { 29 }$
C) $- \frac { \sqrt { 11 } } { 6 }$
D) $\frac { \sqrt { 33 } } { 4 }$

Question

Quizplus · Accepted Answer

In quadrant IV, $\cos \theta > 0$ and $\csc \theta < 0$. Given $\csc \theta = -\frac{5}{3}$, we find $\sin \theta = -\frac{3}{5}$ (since $\csc \theta = \frac{1}{\sin \theta}$). Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we get $\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(-\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$, which is positive in quadrant IV.

Complete the Sentence So the Result Is an Identity cos⁡θ\cos \thetacosθ

Complete the Sentence So the Result Is an Identity $\cos \theta$