Question

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

Explanation:

Step1: Find adjacent side and hypotenuse

Let the adjacent side to angle $\theta$ be $a$, the opposite side be $o = 4$, and the hypotenuse be $h = 7$. Using the Pythagorean theorem $a=\sqrt{h^{2}-o^{2}}=\sqrt{7^{2}-4^{2}}=\sqrt{49 - 16}=\sqrt{33}$.

Step2: Recall trig - function definitions

The cosine of an angle in a right - triangle is $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, so $\cos\theta=\frac{\sqrt{33}}{7}$.

Step3: Recall secant definition

The secant of an angle is $\sec\theta=\frac{1}{\cos\theta}$, so $\sec\theta=\frac{7}{\sqrt{33}}=\frac{7\sqrt{33}}{33}$.

Step4: Recall cotangent definition

The cotangent of an angle is $\cot\theta=\frac{\text{adjacent}}{\text{opposite}}$, so $\cot\theta=\frac{\sqrt{33}}{4}$.

Answer:

$\sec\theta=\frac{7\sqrt{33}}{33}$
$\cot\theta=\frac{\sqrt{33}}{4}$
$\cos\theta=\frac{\sqrt{33}}{7}$