Find the exact value. \( \sin \left(2 \cos ^{-1}\left(\frac{11}{61}\right)\right)=\square \)

To find the exact value of \( \sin\left(2 \cos^{-1}\left(\frac{11}{61}\right)\right) \), we can use the double angle identity for sine: \[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \] Let \(\theta = \cos^{-1}\left(\frac{11}{61}\right)\). This means that \(\cos(\theta) = \frac{11}{61}\). Next, we need to find \(\sin(\theta)\). Using the Pythagorean identity: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] we can rearrange this to find \(\sin(\theta)\): \[ \sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \left(\frac{11}{61}\right)^2 \] Calculating \(\cos^2(\theta)\): \[ \cos^2(\theta) = \frac{121}{3721} \] Thus, \[ \sin^2(\theta) = 1 - \frac{121}{3721} = \frac{3721 - 121}{3721} = \frac{3600}{3721} \] Now, taking the square root, \[ \sin(\theta) = \sqrt{\frac{3600}{3721}} = \frac{60}{61} \] Now we have both \(\sin(\theta)\) and \(\cos(\theta)\): \(\sin(\theta) = \frac{60}{61}\) and \(\cos(\theta) = \frac{11}{61}\). Now we can apply the double angle formula: \[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2 \left(\frac{60}{61}\right) \left(\frac{11}{61}\right) \] Calculating this gives: \[ \sin(2\theta) = 2 \cdot \frac{60 \cdot 11}{61 \cdot 61} = \frac{1320}{3721} \] Thus, the exact value is: \[ \sin\left(2 \cos^{-1}\left(\frac{11}{61}\right)\right) = \frac{1320}{3721} \]