Answer
$$\sin\theta=-\frac{5}{\sqrt{26}}$$
Work Step by Step
$$\cot\theta=-\frac{1}{5}$$
To find $\sin\theta$, first we need to find $\csc\theta$ according to the following identity $$\csc^2\theta=1+\cot^2\theta$$
Apply $\cot\theta=-\frac{1}{5}$ here, we have $$\csc^2\theta=1+(-\frac{1}{5})^2=1+\frac{1}{25}=\frac{26}{25}$$ $$\csc\theta=\pm\frac{\sqrt{26}}{5}$$
We know that $\theta$ is in quadrant IV, which means $\csc\theta$ is negative.
Therefore, $$\csc\theta=-\frac{\sqrt{26}}{5}$$
Now we can find $\sin\theta$ according to the identity $$\csc\theta=\frac{1}{\sin\theta}$$ $$\sin\theta=\frac{1}{\csc\theta}=\frac{1}{-\frac{\sqrt{26}}{5}}=-\frac{5}{\sqrt{26}}$$
