Chapter 4.8, Problem 57E

To determine

To Find: The exact values of the six trigonometric functions of the angle $\theta$

Answer to Problem 57E

The six trigonometric functions of $\theta$ are

  $\sin \theta =\frac{40}{41}$

  $\cos \theta =\frac{9}{41}$

  $\tan \theta =\frac{40}{9}$

  $\cos ec\theta =\frac{41}{40}$

  $\sec \theta =\frac{41}{9}$

  $\cot \theta =\frac{9}{40}$

Explanation of Solution

Given information: In a right-angle triangle

  $\cot \theta =\frac{9}{40}$

Concept used: Let $\theta$ be an acute angle of a right triangle. The six trigonometric functions of the angle $\theta$ are defined as follows

  $\sin \theta =\frac{\text{opp}}{\text{hyp}}$

  $\cos \theta =\frac{\text{adj}}{\text{hyp}}$

  $\tan \theta =\frac{\text{opp}}{\text{adj}}$

  $\cos ec\theta =\frac{\text{hyp}}{\text{opp}}$

  $\sec \theta =\frac{\text{hyp}}{\text{adj}}$

  $\cot \theta =\frac{\text{adj}}{\text{opp}}$

The abbreviations opposite, adjacent, and hypotenuse represent the length of the three sides of a right triangle

  $\text{opp =}$ the length of the side opposite $\theta$

  $\text{adj =}$ the length of the side adjacent to $\theta$

  $\text{hyp =}$ the length of the hypotenuse

Calculation:

Use definition of cotangent function,

  $\begin{array}{l}\cot \theta =\frac{\text{adj}}{\text{opp}}\\ \cot \theta =\frac{9}{40}\end{array}$

Use the Pythagorean theorem

  ${\left(\text{hyp}\right)}^{\text{2}}\text{=}{\left(\text{adj}\right)}^{\text{2}}\text{+}{\left(\text{opp}\right)}^{\text{2}}$

It follows that

  $\text{hyp}=\sqrt{{(9)}^2+{(40)}^2}=41$

So, the six trigonometric functions of $\theta$ are

  $\sin \theta =\frac{40}{41}$

  $\cos \theta =\frac{9}{41}$

  $\tan \theta =\frac{40}{9}$

  $\cos ec\theta =\frac{41}{40}$

  $\sec \theta =\frac{41}{9}$

  $\cot \theta =\frac{9}{40}$