Question

use a sketch to find the exact value of the following expression. cossin^(-1)(-15/17). which of the following triangles can be used to find the exact value of the given expression? cossin^(-1)(-15/17) = □ (type an integer or a simplified fraction. rationalize all denominators.)

Explanation:

Step1: Let $\theta=\sin^{-1}(-\frac{15}{17})$.

This means $\sin\theta =-\frac{15}{17}$, and we consider a right - triangle in the coordinate plane. In a right - triangle, $\sin\theta=\frac{y}{r}$, where $y$ is the vertical side and $r$ is the hypotenuse. So, $y = - 15$ and $r = 17$.

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

We have $x^{2}+(-15)^{2}=17^{2}$, so $x^{2}+225 = 289$, then $x^{2}=289 - 225=64$, and $x = 8$ (we take the positive value of $x$ since for the inverse - sine function, the range of $\sin^{-1}u$ is $[-\frac{\pi}{2},\frac{\pi}{2}]$ and in this range, the cosine of the resulting angle is non - negative).

Step3: Find $\cos\theta$.

Since $\cos\theta=\frac{x}{r}$, and $x = 8$, $r = 17$, then $\cos\theta=\frac{8}{17}$. So, $\cos[\sin^{-1}(-\frac{15}{17})]=\frac{8}{17}$.

Answer:

$\frac{8}{17}$