Question

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

Explanation:

Step1: Identify the sides relative to θ

In the right triangle, for angle θ, the opposite side (opp) is 15, the adjacent side (adj) is 8, and the hypotenuse (hyp) can be found using the Pythagorean theorem: \( hyp = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \).

Step2: Calculate \(\cot\theta\)

The cotangent of an angle in a right triangle is \(\frac{\text{adjacent}}{\text{opposite}}\). So, \(\cot\theta = \frac{8}{15}\).

Step3: Calculate \(\cos\theta\)

The cosine of an angle in a right triangle is \(\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos\theta = \frac{8}{17}\).

Step4: Calculate \(\sec\theta\)

The secant of an angle is the reciprocal of cosine, so \(\sec\theta = \frac{1}{\cos\theta} = \frac{17}{8}\).

Answer:

\(\cot\theta = \frac{8}{15}\), \(\cos\theta = \frac{8}{17}\), \(\sec\theta = \frac{17}{8}\)