Chapter 6.2, Problem 23E

To determine

To calculate: The value of trigonometric ratios if $\sec \theta =\frac{7}{2}$ and also sketch the triangle with an acute angle $\theta$ .

Answer to Problem 23E

The value of trigonometric ratios are,

  $\sin \theta =\frac{3\sqrt{5}}{7},cos\theta =\frac{2}{7},\tan \theta =\frac{3\sqrt{5}}{2},\csc \theta =\frac{7\sqrt{5}}{15},\sec \theta =\frac{7}{2}\text{ and }\cot \theta =\frac{2\sqrt{5}}{15}$ .The sketch of the triangle is,

Explanation of Solution

Given information:

The value of trigonometric function $\sec \theta =\frac{7}{2}$ .

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 provided value of trigonometric function $\sec \theta =\frac{7}{2}$ .

Since, the secant function is expressed as $\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ . Construct a right angle triangle with acute angle $\theta$ and whose adjacent side is 2 units and hypotenuse is $7$ unit.

The figure obtained is provided below,

Observe that adjacent side is 2 units and hypotenuse is $7$ unit.

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

  $\begin{array}{c}{x}^2+2^2=7^2\\ x^2=49-4\\ x^2=45\\ x=3\sqrt{5}\end{array}$

Therefore, length of opposite side is $3\sqrt{5}$ 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,

  $\begin{array}{c}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ =\frac{3\sqrt{5}}{7}\end{array}$

The value of cosine function is,

  $\begin{array}{c}cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ =\frac{2}{7}\end{array}$

The value of tangent function is,

  $\begin{array}{c}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\\ =\frac{3\sqrt{5}}{2}\end{array}$

The value of cosecant function is,

  $\begin{array}{c}\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}}\\ =\frac{7}{3\sqrt{5}}\\ =\frac{7\sqrt{5}}{15}\end{array}$

The value of secant function is,

  $\begin{array}{c}\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}\\ =\frac{7}{2}\end{array}$

The value of cotangent function is,

  $\begin{array}{c}\cot \theta =\frac{\text{adjacent}}{\text{opposite}}\\ =\frac{2}{3\sqrt{5}}\\ =\frac{2\sqrt{5}}{15}\end{array}$

Hence, the value of trigonometric ratios are,

  $\sin \theta =\frac{3\sqrt{5}}{7},cos\theta =\frac{2}{7},\tan \theta =\frac{3\sqrt{5}}{2},\csc \theta =\frac{7\sqrt{5}}{15},\sec \theta =\frac{7}{2}\text{ and }\cot \theta =\frac{2\sqrt{5}}{15}$ .