Question

find $\cos \theta$, $\cot \theta$, and $\sec \theta$, where $\theta$ is the angle shown in the figure. give exact values, not decimal approximations.
$\cos \theta=\square$
$\cot \theta=\square$
$\sec \theta=\square$

Explanation:

Step1: Find the hypotenuse

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 3\), \(b = 4\). Then \(c=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\).

Step2: Calculate \(\cos\theta\)

By the definition \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). The adjacent side to \(\theta\) is \(4\), and the hypotenuse is \(5\). So \(\cos\theta=\frac{4}{5}\).

Step3: Calculate \(\cot\theta\)

By the definition \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\). The adjacent side to \(\theta\) is \(4\), and the opposite side is \(3\). So \(\cot\theta=\frac{4}{3}\).

Step4: Calculate \(\sec\theta\)

By the definition \(\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}\). The hypotenuse is \(5\), and the adjacent side to \(\theta\) is \(4\). So \(\sec\theta=\frac{5}{4}\).

Answer:

\(\cos\theta=\frac{4}{5}\), \(\cot\theta=\frac{4}{3}\), \(\sec\theta=\frac{5}{4}\)