Chapter 4.3, Problem 7E

To determine

Tocalculate : The exact values of the six trigonometric functions of the angle $\theta$ from the given figure.

Answer to Problem 7E

  $\sin \theta =\frac{9}{41},\cos \theta =\frac{40}{41},\tan \theta =\frac{9}{40},\cot \theta =\frac{40}{9},\csc \theta =\frac{41}{9},\sec \theta =\frac{41}{40}$

Explanation of Solution

Given information :

The figure with the opposite side and hypotenuse are $9$ and $41$ units respectively as given below.

  

Formula used :

Pythagorean Theorem: It states that the square of the hypotenuse $\left(\text{hyp}\right)$ is equal to the sum of the square of the adjacent $\left(\text{adj}\right)$ and opposite $\left(\text{opp}\right)$ sides. Mathematically,

  ${\text{hyp}}^2={\text{adj}}^2+{\text{opp}}^2$

Use the formulas for six trigonometric functions written below.

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}},\cos \theta =\frac{\text{adj}}{\text{hyp}},\tan \theta =\frac{\text{opp}}{\text{adj}}\\ \cot \theta =\frac{\text{adj}}{\text{opp}},\csc \theta =\frac{\text{hyp}}{\text{opp}},\sec \theta =\frac{\text{hyp}}{\text{adj}}\end{array}$

Calculation :

Substitute 41 for $\left(\text{hyp}\right)$ and 9 for $\left(\text{opp}\right)$ in the Pythagoraus formula.

  $\begin{array}{c}{\text{hyp}}^2={\text{adj}}^2+{\text{opp}}^2\\ {41}^2={\text{adj}}^2+9^2\\ 1681={\text{adj}}^2+81\end{array}$

Simplify further,

  $\begin{array}{c}{\text{adj}}^2=1681-81\\ \text{adj}=\sqrt{1600}\\ =40\end{array}$

So, the all the six trigonometric functions can be calculated as below:

Substitue 41 for $\left(\text{hyp}\right)$ , 40 for $\left(\text{adj}\right)$ and 9 for $\left(\text{opp}\right)$ in the formula for all the six trigonometric function.

  $\begin{array}{c}\sin \theta =\frac{\text{opp}}{\text{hyp}}\\ =\frac{9}{41}\\ \cos \theta =\frac{\text{adj}}{\text{hyp}}\\ =\frac{40}{41}\end{array}$

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

  $\begin{array}{c}\csc \theta =\frac{\text{hyp}}{\text{opp}}\\ =\frac{41}{9}\\ \sec \theta =\frac{\text{hyp}}{\text{adj}}\\ =\frac{41}{40}\end{array}$