use a sketch to find the exact value of the following expression. cos(sin^(-1)(1/8)) cos(sin^(-1)(1/8)) = □…

Explanation:

Step1: Let $\theta=\sin^{-1}\frac{1}{8}$

By the definition of the inverse - sine function, $\sin\theta=\frac{1}{8}$, and $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$. Consider a right - triangle where the opposite side to the angle $\theta$ is $a = 1$ and the hypotenuse is $c = 8$.

Step2: Use the Pythagorean theorem to find the adjacent side

According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 1$ and $c = 8$. Then $b^{2}=c^{2}-a^{2}$. Substituting the values, we get $b^{2}=8^{2}-1^{2}=64 - 1=63$, so $b=\sqrt{63}=3\sqrt{7}$ (since $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$, the adjacent side $b>0$).

Step3: Find $\cos(\sin^{-1}\frac{1}{8})$

We know that $\cos\theta=\frac{b}{c}$. Since $\theta = \sin^{-1}\frac{1}{8}$, and from the right - triangle $b = 3\sqrt{7}$ and $c = 8$, then $\cos(\sin^{-1}\frac{1}{8})=\frac{3\sqrt{7}}{8}$.

Answer:

$\frac{3\sqrt{7}}{8}$