Question
Let $\theta$ be an acute angle. If $\cos \theta = \frac{3}{11},$ then what is $\sin \theta?$
Answers
GPT 3.5
Since $\cos^2 \theta + \sin^2 \theta = 1,$
\[\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left( \frac{3}{11} \right)^2 = \frac{121 - 9}{121} = \frac{112}{121}.\]Since $\theta$ is acute, $\sin \theta = \sqrt{\frac{112}{121}} = \boxed{\frac{4 \sqrt{7}}{11}}.$
\[\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left( \frac{3}{11} \right)^2 = \frac{121 - 9}{121} = \frac{112}{121}.\]Since $\theta$ is acute, $\sin \theta = \sqrt{\frac{112}{121}} = \boxed{\frac{4 \sqrt{7}}{11}}.$