Question

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

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

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

Explanation:

Step1: Let $\theta=\tan^{-1}\frac{4}{3}$

By the definition of the inverse - tangent function, if $\theta = \tan^{-1}\frac{4}{3}$, then $\tan\theta=\frac{4}{3}$ and $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$. Using the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$ and $\tan\theta=\frac{\sin\theta}{\cos\theta}$, we can find $\sin\theta$ and $\cos\theta$. Since $\tan\theta=\frac{y}{x}=\frac{4}{3}$ (in the right - triangle with opposite side $y = 4$ and adjacent side $x = 3$), the hypotenuse $r=\sqrt{x^{2}+y^{2}}=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=5$. So, $\sin\theta=\frac{4}{5}$ and $\cos\theta=\frac{3}{5}$.

Step2: Use the double - angle formula for sine

The double - angle formula for sine is $\sin(2\alpha)=2\sin\alpha\cos\alpha$. Here, $\alpha=\theta$, so $\sin(2\tan^{-1}\frac{4}{3})=\sin(2\theta)$.
Substituting $\sin\theta=\frac{4}{5}$ and $\cos\theta=\frac{3}{5}$ into the double - angle formula $\sin(2\theta)=2\sin\theta\cos\theta$, we get:

$$
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$$

Answer:

$\frac{24}{25}$