Find $\sin \theta$ if $\cot \theta = - 4$ and $\cos \theta

Question

Find $\sin \theta$ if $\cot \theta = - 4$ and $\cos \theta < 0$.
A) - 17
B) $\frac { \sqrt { 17 } } { 17 }$
C) $- \frac { 1 } { 4 }$
D) $\frac { \sqrt { 17 } } { 4 }$

Quizplus · Accepted Answer

Given $\cot \theta = -4$, we can write this as $\frac{1}{\tan \theta} = -4$, implying $\tan \theta = -\frac{1}{4}$. Since $\tan \theta = \frac{\sin \theta}{\cos \theta}$, and given $\cos \theta < 0$, we are in either the second or third quadrant where $\sin \theta$ is positive in the second quadrant and negative in the third. The relationship between $\sin$, $\cos$, and $\tan$ can be used with the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Substituting $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and solving for $\sin \theta$ gives $\sin \theta = \pm \frac{\sqrt{17}}{17}$. Considering the sign of $\cos \theta$, and that $\sin \theta$ should be negative in the third quadrant (where $\cos \theta < 0$ and $\tan \theta < 0$), the correct answer is $\frac{\sqrt{17}}{17}$, which matches option B.

Find sin⁡θ\sin \thetasinθ If cot⁡θ=−4\cot \theta = - 4cotθ=−4 And cos⁡θ<0\cos \theta < 0cosθ<0

Find $\sin \theta$ If $\cot \theta = - 4$ And $\cos \theta < 0$