Chapter 4.3, Problem 11E

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 11E

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

  $\begin{array}{l}\sin \theta =\frac{8}{17}\\ \cos \theta =\frac{15}{17}\\ \tan \theta =\frac{8}{15}\\ \cos ec\theta =\frac{17}{8}\\ \sec \theta =\frac{17}{15}\\ \cot \theta =\frac{15}{8}\end{array}$

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:-

  $\begin{array}{l}{\text{Hypotenuse}}^{\text{2}}{\text{=Base}}^{\text{2}}{\text{+Perpendicular}}^{\text{2}}\\ {\text{Hypotenuse}}^{\text{2}}={15}^{\text{2}}+{\text{8}}^{\text{2}}\\ \text{Hypotenuse}=17\end{array}$

We have from the given drawing

  $\begin{array}{l}\sin \theta =\frac{\text{Opposite side}}{\text{Hypotenuse}}=\frac{8}{17}\\ \cos \theta =\frac{\text{Adjacent side}}{\text{Hypotenuse}}=\frac{15}{17}\\ \tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}=\frac{8}{15}\\ \cos ec\theta =\frac{\text{Hypotenuse}}{\text{Opposite side}}=\frac{17}{8}\\ \sec \theta =\frac{\text{Hypotenuse}}{\text{Adjacent side}}=\frac{17}{15}\\ \cot \theta =\frac{\text{Adjacent side}}{\text{Opposite side}}=\frac{15}{8}\end{array}$

For triangle 2:-

  $\begin{array}{l}{\text{Hypotenuse}}^{\text{2}}{\text{=Base}}^{\text{2}}{\text{+Perpendicular}}^{\text{2}}\\ {\text{Hypotenuse}}^{\text{2}}={7.5}^{\text{2}}+{\text{4}}^{\text{2}}\\ \text{Hypotenuse}=8.5\end{array}$

We have from the given drawing

  $\begin{array}{l}\sin \theta =\frac{\text{Opposite side}}{\text{Hypotenuse}}=\frac{4}{8.5}=\frac{8}{17}\\ \cos \theta =\frac{\text{Adjacent side}}{\text{Hypotenuse}}=\frac{7.5}{8.5}=\frac{15}{17}\\ \tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}=\frac{4}{7.5}=\frac{8}{15}\\ \cos ec\theta =\frac{\text{Hypotenuse}}{\text{Opposite side}}=\frac{8.5}{4}=\frac{17}{8}\\ \sec \theta =\frac{\text{Hypotenuse}}{\text{Adjacent side}}=\frac{8.5}{7.5}=\frac{17}{15}\\ \cot \theta =\frac{\text{Adjacent side}}{\text{Opposite side}}=\frac{7.5}{4}=\frac{15}{8}\end{array}$

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

  $\frac{15}{7.5}=\frac{8}{4}=\frac{17}{8.5}=2$

Conclusion:

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

  $\begin{array}{l}\sin \theta =\frac{8}{17}\\ \cos \theta =\frac{15}{17}\\ \tan \theta =\frac{8}{15}\\ \cos ec\theta =\frac{17}{8}\\ \sec \theta =\frac{17}{15}\\ \cot \theta =\frac{15}{8}\end{array}$

The trigonometric functions are same for similar triangles.