Question

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

Explanation:

Step1: Find the adjacent side, opposite side and hypotenuse

Adjacent side $a = 3$, opposite side $b$ (unknown), hypotenuse $c = 8$.

Step2: Use the Pythagorean theorem to find the opposite side

By $a^{2}+b^{2}=c^{2}$, we have $b=\sqrt{c^{2}-a^{2}}=\sqrt{64 - 9}=\sqrt{55}$.

Step3: Calculate $\cos\theta$

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{3}{8}$.

Step4: Calculate $\cot\theta$

$\cot\theta=\frac{\text{adjacent}}{\text{opposite}}=\frac{3}{\sqrt{55}}=\frac{3\sqrt{55}}{55}$.

Step5: Calculate $\sec\theta$

$\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{8}{3}$.

Answer:

$\cos\theta=\frac{3}{8}$
$\cot\theta=\frac{3\sqrt{55}}{55}$
$\sec\theta=\frac{8}{3}$