Question

use a sketch to find the exact value of the following expression. cos(sin^(-1)(1/3)) which of the following sketches is used to find the exact value of the given expression? a. diagram description not included as text - only key elements kept (3,1) r 1 3 x y θ b. (x,1) 3 1 x y θ c. (1,3) r 3 1 x y θ d. (1,y) 3 y 1 x y θ cos(sin^(-1)(1/3)) = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Let $\theta=\sin^{-1}\frac{1}{3}$, then $\sin\theta=\frac{1}{3}$.

In a right - triangle, if $\sin\theta=\frac{y}{r}=\frac{1}{3}$, we can assume $y = 1$ and $r=3$.

Step2: Use the Pythagorean theorem $x^{2}+y^{2}=r^{2}$ to find $x$.

We have $x=\sqrt{r^{2}-y^{2}}=\sqrt{9 - 1}=\sqrt{8}=2\sqrt{2}$.

Step3: Find $\cos\theta$.

Since $\cos\theta=\frac{x}{r}$, and $x = 2\sqrt{2}$, $r = 3$, then $\cos\theta=\frac{2\sqrt{2}}{3}$. So $\cos(\sin^{-1}\frac{1}{3})=\frac{2\sqrt{2}}{3}$.

The correct sketch is B because if $\sin\theta=\frac{1}{3}=\frac{y}{r}$, we have $y = 1$ and $r = 3$, and we need to find $x$ using the right - triangle relationship.

Answer:

B.
$\frac{2\sqrt{2}}{3}$