Question

use a sketch to find the exact value of the following expression.
cos(sin^(-1)5/6)
which of the following sketches is used to find the exact value of the given expression?
a. sketch with point (6,5), r, angle θ, x - axis and y - axis, 6 on x - axis and 5 on y - axis
b. sketch with point (5,6), r, angle θ, x - axis and y - axis, 5 on x - axis and 6 on y - axis
c. sketch with point (5,y), r, angle θ, x - axis and y - axis, 5 on x - axis and 6 on hypotenuse
d. sketch with point (x,5), r, angle θ, x - axis and y - axis, x on x - axis and 5 on y - axis
cos(sin^(-1)5/6)=□
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Let $\theta=\sin^{-1}\frac{5}{6}$.

This means $\sin\theta=\frac{5}{6}$, and in a right - triangle, if $\sin\theta=\frac{y}{r}$, then $y = 5$ and $r=6$.

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

By the Pythagorean theorem $x=\sqrt{r^{2}-y^{2}}$. Substituting $r = 6$ and $y = 5$, we get $x=\sqrt{6^{2}-5^{2}}=\sqrt{36 - 25}=\sqrt{11}$.

Step3: Find $\cos\theta$.

Since $\cos\theta=\frac{x}{r}$, and $x=\sqrt{11}$, $r = 6$, then $\cos\theta=\frac{\sqrt{11}}{6}$. So $\cos(\sin^{-1}\frac{5}{6})=\frac{\sqrt{11}}{6}$.

Answer:

$\frac{\sqrt{11}}{6}$