Question

give the exact value of the expression without using a calculator.

\\( \sin \left(2 \tan ^{-1} \frac{3}{4}\
ight) \\)

\\( \sin \left(2 \tan ^{-1} \frac{3}{4}\
ight)=\square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Let \( \theta = \tan^{-1}\frac{3}{4} \), so \( \tan\theta=\frac{3}{4} \)

We can imagine a right triangle where the opposite side to \( \theta \) is 3 and the adjacent side is 4. Then, by the Pythagorean theorem, the hypotenuse \( r \) is \( \sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25} = 5 \). So, \( \sin\theta=\frac{3}{5} \) and \( \cos\theta=\frac{4}{5} \)

Step2: Use the double - angle formula for sine, \( \sin(2\alpha)=2\sin\alpha\cos\alpha \)

Here, \( \alpha=\theta \), so we substitute \( \sin\theta=\frac{3}{5} \) and \( \cos\theta=\frac{4}{5} \) into the formula.
\( \sin(2\tan^{-1}\frac{3}{4})=\sin(2\theta)=2\sin\theta\cos\theta \)
Substitute the values of \( \sin\theta \) and \( \cos\theta \):
\( 2\times\frac{3}{5}\times\frac{4}{5}=\frac{24}{25} \)

Answer:

\( \frac{24}{25} \)