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

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Calculate or simplify the expression $$ \sin(2 \cos^{-1}(\frac{11}{61})) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sin\left(2\arccos\left(\frac{11}{61}ight)ight)$$
- step1: Rewrite the expression:
  $$2\sin\left(\arccos\left(\frac{11}{61}ight)ight)\cos\left(\arccos\left(\frac{11}{61}ight)ight)$$
- step2: Rewrite the expression:
  $$2\sin\left(\arccos\left(\frac{11}{61}ight)ight)	imes \frac{11}{61}$$
- step3: Rewrite the expression:
  $$2\sqrt{1-\left(\frac{11}{61}ight)^{2}}	imes \frac{11}{61}$$
- step4: Subtract the numbers:
  $$2\sqrt{\frac{3600}{3721}}	imes \frac{11}{61}$$
- step5: Simplify the root:
  $$2	imes \frac{60}{61}	imes \frac{11}{61}$$
- step6: Multiply the terms:
  $$\frac{120}{61}	imes \frac{11}{61}$$
- step7: Multiply the fractions:
  $$\frac{120	imes 11}{61	imes 61}$$
- step8: Multiply:
  $$\frac{1320}{3721}$$
The exact value of $$ \sin \left(2 \cos ^{-1}\left(\frac{11}{61}ight)ight) $$ is $$ \frac{1320}{3721} $$.
The exact value is $$ \frac{1320}{3721} $$.

Find the exact value.

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