evaluate tan(sin^(-1)(-4/5)). enter your answer as a fraction using the slash bar ( /).

Explanation:

Step1: Let $\theta=\sin^{-1}(-\frac{4}{5})$

This means $\sin\theta = -\frac{4}{5}$.

Step2: Determine the quadrant of $\theta$

Since $\sin\theta<0$, $\theta$ is in either the third or fourth - quadrant. The range of $y = \sin^{-1}x$ is $[-\frac{\pi}{2},\frac{\pi}{2}]$, so $\theta$ is in the fourth - quadrant.

Step3: Use the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$

We know $\sin\theta=-\frac{4}{5}$, so $\cos\theta=\sqrt{1 - \sin^{2}\theta}=\sqrt{1-(-\frac{4}{5})^{2}}=\sqrt{1-\frac{16}{25}}=\sqrt{\frac{9}{25}}=\frac{3}{5}$ (positive because $\theta$ is in the fourth - quadrant).

Step4: Calculate $\tan\theta$

Since $\tan\theta=\frac{\sin\theta}{\cos\theta}$, substituting $\sin\theta = -\frac{4}{5}$ and $\cos\theta=\frac{3}{5}$, we get $\tan\theta=\frac{-\frac{4}{5}}{\frac{3}{5}}=-\frac{4}{3}$.

Answer:

$-\frac{4}{3}$