Chapter 4.3, Problem 12E

To determine

The exact values of the six trigonometric functions of the angle $\theta$ for the two triangles and explain why the values are equal.

Answer to Problem 12E

The exact values of the six trigonometric functions of the angle $\theta$ in both triangles are

  $\begin{array}{l}\sin \theta =\frac{3}{5}\\ \cos \theta =\frac{4}{5}\\ \tan \theta =\frac{3}{4}\\ \cos ec\theta =\frac{5}{3}\\ \sec \theta =\frac{5}{4}\\ \cot \theta =\frac{4}{3}\end{array}$

The trigonometric functions are same for similar triangles.

Explanation of Solution

Given info:

  

Formula used:

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

Calculation:

Using Pythagorean theorem

For triangle 1:-

  $\begin{array}{l}{\text{Hypotenuse}}^{\text{2}}{\text{=Base}}^{\text{2}}{\text{+Perpendicular}}^{\text{2}}\\ \text{1}{\text{.25}}^{\text{2}}=1^{\text{2}}+{\text{Perpendicular}}^{\text{2}}\\ \text{Perpendicular}=0.75\end{array}$

We have from the given drawing

  $\begin{array}{l}\sin \theta =\frac{\text{Opposite side}}{\text{Hypotenuse}}=\frac{0.75}{1.25}=\frac{3}{5}\\ \cos \theta =\frac{\text{Adjacent side}}{\text{Hypotenuse}}=\frac{1}{1.25}=\frac{4}{5}\\ \tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}=\frac{0.75}{1}=\frac{3}{4}\\ \cos ec\theta =\frac{\text{Hypotenuse}}{\text{Opposite side}}=\frac{1.25}{0.75}=\frac{5}{3}\\ \sec \theta =\frac{\text{Hypotenuse}}{\text{Adjacent side}}=\frac{1.25}{1}=\frac{5}{4}\\ \cot \theta =\frac{\text{Adjacent side}}{\text{Opposite side}}=\frac{1}{0.75}=\frac{4}{3}\end{array}$

For triangle 2:-

  $\begin{array}{l}{\text{Hypotenuse}}^{\text{2}}{\text{=Base}}^{\text{2}}{\text{+Perpendicular}}^{\text{2}}\\ {\text{5}}^{\text{2}}=4^{\text{2}}+{\text{Perpendicular}}^{\text{2}}\\ \text{Perpendicular}=3\end{array}$

We have from the given drawing

  $\begin{array}{l}\sin \theta =\frac{\text{Opposite side}}{\text{Hypotenuse}}=\frac{3}{5}\\ \cos \theta =\frac{\text{Adjacent side}}{\text{Hypotenuse}}=\frac{4}{5}\\ \tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}=\frac{3}{4}\\ \cos ec\theta =\frac{\text{Hypotenuse}}{\text{Opposite side}}=\frac{5}{3}\\ \sec \theta =\frac{\text{Hypotenuse}}{\text{Adjacent side}}=\frac{5}{4}\\ \cot \theta =\frac{\text{Adjacent side}}{\text{Opposite side}}=\frac{4}{3}\end{array}$

Here the two triangles are similar, hence the values of trigonometric functions are same for both the triangles.

  $\frac{5}{1.25}=\frac{4}{1}=\frac{3}{0.75}=4$

Conclusion:

Thus,the exact values of the six trigonometric functions of the angle $\theta$ in both triangles are

  $\begin{array}{l}\sin \theta =\frac{3}{5}\\ \cos \theta =\frac{4}{5}\\ \tan \theta =\frac{3}{4}\\ \cos ec\theta =\frac{5}{3}\\ \sec \theta =\frac{5}{4}\\ \cot \theta =\frac{4}{3}\end{array}$

The trigonometric functions are same for similar triangles.