Chapter 6.2, Problem 39E

To determine

To calculate: The sides of the triangle and six trigonometric ratios of $\theta$ for the given figure.

  

Answer to Problem 39E

The triangle along with sides is,

  

The six trigonometric ratios of $\theta$ are $\sin \theta =\text{0}\text{.5},cos\theta =0.87,\tan \theta =\text{0}\text{.58},\csc \theta =\text{2},\sec \theta =\text{1}\text{.15}$ and $\text{ cot}\theta \text{ = 1}\text{.7}$

Explanation of Solution

Given information:

The right angle triangle with angle $\theta$ .

  

Formula used:

The trigonometric ratios for a right angle triangle are defined as,

  $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}},\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}},\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ and $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Calculation:

Consider the right angle triangle with length of its sides and angle $\theta$ .

  

Measure the lengths of sides of the triangle using ruler and label the figure as,

  

Recall that the trigonometric ratios for a right angle triangle are defined as,

  $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}},\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}},\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ and $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Apply it, to estimate the value of trigonometric ratios,

The value of sine function is,

  $\begin{array}{c}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ =\frac{1.9}{3.8}\\ =0.5\end{array}$

The value of cosine function is,

  $\begin{array}{c}cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ =\frac{3.3}{3.8}\\ =0.87\end{array}$

The value of tangent function is,

  $\begin{array}{c}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\\ =\frac{1.9}{3.3}\\ =0.58\end{array}$

The value cosecant function is,

  $\begin{array}{c}\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}}\\ =\frac{3.8}{1.9}\\ =2\end{array}$

The value of secant function is,

  $\begin{array}{c}\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}\\ =\frac{3.8}{3.3}\\ =1.15\end{array}$

The value of cotangent function is,

  $\begin{array}{c}\cot \theta =\frac{\text{adjacent}}{\text{opposite}}\\ =\frac{3.3}{1.9}\\ =1.7\end{array}$

Thus, The six trigonometric ratios of $\theta$ are $\sin \theta =\text{0}\text{.5},cos\theta =0.87,\tan \theta =\text{0}\text{.58},\csc \theta =\text{2},\sec \theta =\text{1}\text{.15}$ and $\text{ cot}\theta \text{ = 1}\text{.7}$