Find $\sin \theta$.
\[
\cot \theta=-\frac{2}{7}, \theta \text { in quadrant IV }
\]
\[
\sin \theta=
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expressions.)
Answer
So, the final answer is \(\boxed{-\frac{7\sqrt{53}}{53}}\).
Step 1 ：Since \(\cot \theta = -\frac{2}{7}\), we know that \(\cos \theta = -\frac{2}{\sqrt{2^2 + 7^2}} = -\frac{2}{\sqrt{53}}\) and \(\sin \theta = \frac{7}{\sqrt{2^2 + 7^2}} = \frac{7}{\sqrt{53}}\).
Step 2 ：However, since \(\theta\) is in the fourth quadrant where sine is negative, we must take the negative of the above value for \(\sin \theta\).
Step 3 ：So, \(\sin \theta = -\frac{7}{\sqrt{53}}\).
Step 4 ：We can rationalize the denominator to get \(\sin \theta = -\frac{7\sqrt{53}}{53}\).
Step 5 ：So, the final answer is \(\boxed{-\frac{7\sqrt{53}}{53}}\).
