Find the exact value of $$ an \left[2 \cos ^{-1}\left(-\frac{3}{5} ight) ight] $$ $$ an \left[2 \cos ^{-1}\left(-\frac{3}{5} ight) ight]=\square $$ (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

Question

Find the exact value of $$ 	an \left[2 \cos ^{-1}\left(-\frac{3}{5}ight)ight] $$ $$ 	an \left[2 \cos ^{-1}\left(-\frac{3}{5}ight)ight]=\square $$ (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

UpStudy AI · Accepted Answer

Find the value of $$ 	an(2 \cos^{-1}(-\frac{3}{5})) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$	an\left(2\arccos\left(-\frac{3}{5}ight)ight)$$
- step1: Rewrite the expression:
  $$\frac{2	an\left(\arccos\left(-\frac{3}{5}ight)ight)}{1-	an^{2}\left(\arccos\left(-\frac{3}{5}ight)ight)}$$
- step2: Rewrite the expression:
  $$\frac{2	imes \frac{\sqrt{1-\left(-\frac{3}{5}ight)^{2}}}{-\frac{3}{5}}}{1-	an^{2}\left(\arccos\left(-\frac{3}{5}ight)ight)}$$
- step3: Rewrite the expression:
  $$\frac{2	imes \frac{\sqrt{1-\left(-\frac{3}{5}ight)^{2}}}{-\frac{3}{5}}}{1-\left(\frac{\sqrt{1-\left(-\frac{3}{5}ight)^{2}}}{-\frac{3}{5}}ight)^{2}}$$
- step4: Subtract the numbers:
  $$\frac{2	imes \frac{\sqrt{1-\left(-\frac{3}{5}ight)^{2}}}{-\frac{3}{5}}}{1-\left(\frac{\sqrt{\frac{16}{25}}}{-\frac{3}{5}}ight)^{2}}$$
- step5: Subtract the numbers:
  $$\frac{2	imes \frac{\sqrt{\frac{16}{25}}}{-\frac{3}{5}}}{1-\left(\frac{\sqrt{\frac{16}{25}}}{-\frac{3}{5}}ight)^{2}}$$
- step6: Simplify the root:
  $$\frac{2	imes \frac{\sqrt{\frac{16}{25}}}{-\frac{3}{5}}}{1-\left(\frac{\frac{4}{5}}{-\frac{3}{5}}ight)^{2}}$$
- step7: Simplify the root:
  $$\frac{2	imes \frac{\frac{4}{5}}{-\frac{3}{5}}}{1-\left(\frac{\frac{4}{5}}{-\frac{3}{5}}ight)^{2}}$$
- step8: Divide the terms:
  $$\frac{2	imes \frac{\frac{4}{5}}{-\frac{3}{5}}}{1-\left(-\frac{4}{3}ight)^{2}}$$
- step9: Divide the terms:
  $$\frac{2\left(-\frac{4}{3}ight)}{1-\left(-\frac{4}{3}ight)^{2}}$$
- step10: Multiply the numbers:
  $$\frac{-\frac{8}{3}}{1-\left(-\frac{4}{3}ight)^{2}}$$
- step11: Subtract the numbers:
  $$\frac{-\frac{8}{3}}{-\frac{7}{9}}$$
- step12: Multiply by the reciprocal:
  $$-\frac{8}{3}\left(-\frac{9}{7}ight)$$
- step13: Use the rules for multiplication and division:
  $$\frac{8}{3}	imes \frac{9}{7}$$
- step14: Reduce the numbers:
  $$8	imes \frac{3}{7}$$
- step15: Multiply:
  $$\frac{8	imes 3}{7}$$
- step16: Multiply:
  $$\frac{24}{7}$$
The exact value of $$ 	an \left[2 \cos ^{-1}\left(-\frac{3}{5}ight)ight] $$ is $$ \frac{24}{7} $$ or $$ 3\frac{3}{7} $$ or $$ 3.4\overline{2857} $$.
$$ 	an \left[2 \cos^{-1}\left(-\frac{3}{5}ight)ight] = \frac{24}{7} $$

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