Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\cot \theta = - 2$ and $\cos \theta

Question

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\cot \theta = - 2$ and $\cos \theta < 0$.
A) $\frac { \sqrt { 5 } } { 5 }$
B) $\frac { \sqrt { 5 } } { 2 }$
C) $- \frac { 1 } { 2 }$
D) $- 5$

Quizplus · Accepted Answer

Given $\cot \theta = -2$, we can express this in terms of sine and cosine as $\frac{\cos \theta}{\sin \theta} = -2$. Since $\cos \theta < 0$, we are in either the second or third quadrant, where sine is positive in the second quadrant and negative in the third. The negative cotangent value suggests we are in the third quadrant (where both sine and cosine are negative), but we need to find the sine value explicitly. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, and knowing $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$, we can find $\sin \theta$ and $\cos \theta$. If $\cot \theta = -2$, then $\tan \theta = -\frac{1}{2}$, and we can set up a right triangle (or use identities) to find that if $\cos \theta = -\frac{2}{\sqrt{5}}$, then $\sin \theta = -\frac{1}{\sqrt{5}}$. However, since we are looking for the positive or negative value based on the quadrant, and the question specifies $\cos \theta < 0$, we deduce that $\sin \theta$ must be negative in the context of the given information, leading to a contradiction in my explanation regarding the sign based on the quadrant. Correcting this oversight, the focus should be on the calculation of $\sin \theta$ directly from the given $\cot \theta$, which involves understanding that $\sin^2 \theta + \cos^2 \theta = 1$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$ imply a relationship that can be solved for $\sin \theta$ without making assumptions about the specific quadrant beyond the sign of $\cos \theta$. Given $\cot \theta = -2$, we find that $\sin \theta$ and $\cos \theta$ must satisfy the Pythagorean identity, leading to $\sin \theta = \pm\frac{\sqrt{5}}{5}$, with the sign determined by the specific quadrant information, which was initially misapplied. Correctly, without assuming the quadrant directly, we should solve for $\sin \theta$ using the identity and given information about $\cot \theta$, leading to a calculation error in determining the exact value of $\sin \theta$ based on the provided $\cot \theta$. The correct approach involves using the relationship between $\cot \theta$, $\sin \theta$, and $\cos \theta$ to directly calculate $\sin \theta$, considering the Pythagorean identity and the given condition on $\cos \theta$'s sign.

Use the Fundamental Identities to Find the Value of the Trigonometric