Chapter 4.3, Problem 21E

To determine

The sketch of a right triangle corresponding to the trigonometric function of the acute angle $\theta$ , then find the remaining 5 trigonometric function values.

Answer to Problem 21E

The remaining 5 trigonometric function values of the angle $\theta$ are

  $\begin{array}{l}\sin \theta =\frac{1}{3.16}\\ \cos \theta =\frac{3}{3.16}\\ \tan \theta =\frac{1}{3}\\ \cos ec\theta =\frac{3.16}{1}\\ sec\theta =\frac{3.16}{3}\end{array}$

Explanation of Solution

Given info:

  $\cot \theta =3$

Formula used:

  ${\text{Hypotenuse}}^{\text{2}}{\text{=Base}}^{\text{2}}{\text{+Perpendicular}}^{\text{2}}$

Calculation:

The right-angled triangle is

  

Using Pythagorean theorem

  $\begin{array}{l}{\text{Hypotenuse}}^{\text{2}}{\text{=Base}}^{\text{2}}{\text{+Perpendicular}}^{\text{2}}\\ {\text{Hypotenuse}}^{\text{2}}={\text{3}}^{\text{2}}+{\text{1}}^{\text{2}}\\ \text{Hypotenuse}=3.16\end{array}$

We have from the given drawing

  $\begin{array}{l}\sin \theta =\frac{\text{Opposite side}}{\text{Hypotenuse}}=\frac{1}{3.16}\\ \cos \theta =\frac{\text{Adjacent side}}{\text{Hypotenuse}}=\frac{3}{3.16}\\ \tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}=\frac{1}{3}\\ \cos ec\theta =\frac{\text{Hypotenuse}}{\text{Opposite side}}=\frac{3.16}{1}\\ sec\theta =\frac{\text{Hypotenuse}}{\text{Adjacent side}}=\frac{3.16}{3}\end{array}$

Conclusion:

Thus,the remaining 5 trigonometric function values of the angle $\theta$ are

  $\begin{array}{l}\sin \theta =\frac{1}{3.16}\\ \cos \theta =\frac{3}{3.16}\\ \tan \theta =\frac{1}{3}\\ \cos ec\theta =\frac{3.16}{1}\\ sec\theta =\frac{3.16}{3}\end{array}$