evaluate the expression without using a calculator. cos(sin^(-1)(4/5))

Explanation:

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

By the definition of inverse - sine function, $\sin\theta=\frac{4}{5}$, and $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$.

Step2: Use the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$

We know that $\cos^{2}\theta=1 - \sin^{2}\theta$. Substitute $\sin\theta=\frac{4}{5}$ into the identity: $\cos^{2}\theta=1-(\frac{4}{5})^{2}=1-\frac{16}{25}=\frac{9}{25}$.

Step3: Determine the sign of $\cos\theta$

Since $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ and $\sin\theta=\frac{4}{5}>0$, $\theta$ is in the first - quadrant where $\cos\theta>0$. So, $\cos\theta=\sqrt{\frac{9}{25}}=\frac{3}{5}$.

Answer:

$\frac{3}{5}$