Question

find cotθ, cscθ, and sinθ, where θ is the angle shown in the figure. give exact values, not decimal approximations.

Explanation:

Step1: Find the adjacent side using Pythagorean theorem

Let the adjacent side be $a$. By the Pythagorean theorem $a=\sqrt{6^{2}-5^{2}}=\sqrt{36 - 25}=\sqrt{11}$.

Step2: Recall the definitions of trigonometric functions

The sine of an angle $\theta$ in a right - triangle is $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\sin\theta=\frac{5}{6}$.
The cosecant of an angle $\theta$ is $\csc\theta=\frac{1}{\sin\theta}$, so $\csc\theta=\frac{6}{5}$.
The cotangent of an angle $\theta$ is $\cot\theta=\frac{\text{adjacent}}{\text{opposite}}$, so $\cot\theta=\frac{\sqrt{11}}{5}$.

Answer:

$\cot\theta=\frac{\sqrt{11}}{5}$, $\csc\theta=\frac{6}{5}$, $\sin\theta=\frac{5}{6}$