Give the exact value of the expression without using a calculator.
$\sin (2 \tan^{- 1} \frac{15}{8})$
$\sin (2 \tan^{- 1} \frac{15}{8}) =$
(Use radicals or fractions for any numbers in the expression.)

Answer

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Answer

Simplify the fraction to get the final answer: $- \frac{1932}{5216.445}$ .

Steps

Step 1 ：Let $a = 2 \tan^{- 1} \frac{15}{8}$ . Then $\tan a = \tan (2 \tan^{- 1} \frac{15}{8})$ .

Step 2 ：Using the double angle formula for tangent, we have $\tan a = \frac{2 \cdot \frac{15}{8}}{1 - (\frac{15}{8})^{2}} = \frac{30}{64 - 225} = - \frac{30}{161}$ .

Step 3 ：Construct a right triangle with opposite side -30 and adjacent side 161. The hypotenuse is $\sqrt{(- 30)^{2} + 161^{2}} = \sqrt{26081} = 161.5$ .

Step 4 ：Then, $\sin a = \frac{- 30}{161.5} = - \frac{6}{32.3}$ and $\cos a = \frac{161}{161.5} = \frac{322}{322.3}$ .

Step 5 ：Using the double angle formula for sine, we have $\sin 2 a = 2 \sin a \cos a = 2 \cdot - \frac{6}{32.3} \cdot \frac{322}{322.3} = - \frac{3864}{10432.89}$ .

Step 6 ：Simplify the fraction to get the final answer: $- \frac{1932}{5216.445}$ .

Give the exact value of the expression without using a calculator.sin⁡(2tan−1⁡158)sin⁡(2tan−1⁡158)=(Use radicals or fractions for any numbers in the expression.)

Answer

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Give the exact value of the expression without using a calculator.
$\sin (2 \tan^{- 1} \frac{15}{8})$
$\sin (2 \tan^{- 1} \frac{15}{8}) =$
(Use radicals or fractions for any numbers in the expression.)