Question

use a sketch to find the exact value of the following expression.
cot(sin^(-1)12/13)
which of the following triangles can be used to find the exact value of the given expression?
a. image of a right - triangle with r = 13, y = 12
b. image of a right - triangle with x = 12, y = 13
c. image of a right - triangle with r = 13, x = 12
d. image of a right - triangle with x = 13, y = 12
cot(sin^(-1)12/13)=□ (type an integer or a simplified fraction. rationalize all denominators.)

Explanation:

Step1: Recall the definition of sine

Let $\theta=\sin^{-1}\frac{12}{13}$, then $\sin\theta=\frac{y}{r}=\frac{12}{13}$, where $y = 12$ and $r=13$. In a right - triangle, by the Pythagorean theorem $x=\sqrt{r^{2}-y^{2}}$.

Step2: Calculate the value of $x$

$x=\sqrt{13^{2}-12^{2}}=\sqrt{169 - 144}=\sqrt{25}=5$.

Step3: Recall the definition of cotangent

$\cot\theta=\frac{x}{y}$. Since $\theta = \sin^{-1}\frac{12}{13}$, and we found $x = 5$ and $y = 12$, then $\cot(\sin^{-1}\frac{12}{13})=\frac{5}{12}$.

Answer:

$\frac{5}{12}$