Question

use a sketch to find the exact value of the following expression.
\\(\cot(\sin^{- 1}\frac{60}{61})\\)
which of the following triangles can be used to find the exact value of the given expression?
a.
b.
c.
d.
\\(\cot(\sin^{- 1}\frac{60}{61})=\\)
(type an integer or a simplified fraction. rationalize all denominators.)

Explanation:

Step1: Recall the definition of inverse - sine

Let $\theta=\sin^{-1}\frac{60}{61}$. Then $\sin\theta=\frac{60}{61}$, and in a right - triangle, $\sin\theta=\frac{y}{r}$, where $y$ is the length of the opposite side and $r$ is the length of the hypotenuse. So, $y = 60$ and $r = 61$.

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

By the Pythagorean theorem $x=\sqrt{r^{2}-y^{2}}$. Substitute $r = 61$ and $y = 60$ into the formula: $x=\sqrt{61^{2}-60^{2}}=\sqrt{(61 + 60)(61 - 60)}=\sqrt{121\times1}=11$.

Step3: Recall the definition of cotangent

$\cot\theta=\frac{x}{y}$. Since $\theta=\sin^{-1}\frac{60}{61}$, and we found $x = 11$ and $y = 60$, then $\cot(\sin^{-1}\frac{60}{61})=\frac{11}{60}$.

Answer:

$\frac{11}{60}$