Chapter 6.2, Problem 8E

To determine

To calculate: The value of trigonometric ratios of the angle $\theta$ provided in the triangle.

Answer to Problem 8E

The value of trigonometric ratios are,

  $\sin \theta =\frac{7}{8},cos\theta =\frac{\sqrt{15}}{8},\tan \theta =\frac{7\sqrt{15}}{15},\csc \theta =\frac{8}{7},\sec \theta =\frac{8\sqrt{15}}{15}\text{ and }\cot \theta =\frac{\sqrt{15}}{7}$ .

Explanation of Solution

Given information:

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

Observe that opposite side is of length 7 units and hypotenuse has length 8 units.

Now, let the length of adjacent side be x , as it is a right angle triangle so,

  $\begin{array}{c}{x}^2+7^2=8^2\\ x^2=64-49\\ x^2=15\\ x=\sqrt{15}\end{array}$

Therefore, length of adjacent side is $\sqrt{15}$ units.

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,

  $\begin{array}{c}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ =\frac{7}{8}\end{array}$

The value of cosine function is,

  $\begin{array}{c}cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ =\frac{\sqrt{15}}{8}\end{array}$

The value of tangent function is,

  $\begin{array}{c}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\\ =\frac{7}{\sqrt{15}}\\ =\frac{7\sqrt{15}}{15}\end{array}$

The value of cosecant function is,

  $\begin{array}{c}\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}}\\ =\frac{8}{7}\end{array}$

The value of secant function is,

  $\begin{array}{c}\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}\\ =\frac{8}{\sqrt{15}}\\ =\frac{8\sqrt{15}}{15}\end{array}$

The value of cotangent function is,

  $\begin{array}{c}\cot \theta =\frac{\text{adjacent}}{\text{opposite}}\\ =\frac{\sqrt{15}}{7}\end{array}$

Hence, the value of trigonometric ratios are,

  $\sin \theta =\frac{7}{8},cos\theta =\frac{\sqrt{15}}{8},\tan \theta =\frac{7\sqrt{15}}{15},\csc \theta =\frac{8}{7},\sec \theta =\frac{8\sqrt{15}}{15}\text{ and }\cot \theta =\frac{\sqrt{15}}{7}$ .