Question

use a sketch to find the exact value of the following expression.
cos(sin^(-1)(8/10))
which of the following triangles is used to find the exact value of the given expression?
cos(sin^(-1)(8/10)) = (type an integer or a simplified fraction.)

Explanation:

Step1: Recall the definition of inverse - sine.

Let $\theta=\sin^{-1}\frac{8}{10}$. Then $\sin\theta=\frac{8}{10}=\frac{y}{r}$, where $y = 8$ and $r = 10$ in a right - triangle ($r$ is the hypotenuse, $y$ is the opposite side to the angle $\theta$).

Step2: Use the Pythagorean theorem to find the adjacent side $x$.

By the Pythagorean theorem $x=\sqrt{r^{2}-y^{2}}$. Substitute $r = 10$ and $y = 8$ into the formula: $x=\sqrt{10^{2}-8^{2}}=\sqrt{100 - 64}=\sqrt{36}=6$.

Step3: Recall the definition of cosine.

We know that $\cos\theta=\frac{x}{r}$. Since $x = 6$ and $r = 10$, then $\cos(\sin^{-1}\frac{8}{10})=\cos\theta=\frac{6}{10}=\frac{3}{5}$.

Answer:

$\frac{3}{5}$