Chapter 6.2, Problem 22E

To determine

To calculate: The value of trigonometric ratios if $\tan \theta =\sqrt{3}$ and also sketch the triangle with an acute angle $\theta$ .

Answer to Problem 22E

The value of trigonometric ratios are,

  $\sin \theta =\frac{\sqrt{3}}{2},cos\theta =\frac{1}{2},\tan \theta =\sqrt{3},\csc \theta =\frac{2\sqrt{3}}{3},\sec \theta =2\text{ and }\cot \theta =\frac{\sqrt{3}}{3}$ .The sketch of the triangle is,

Explanation of Solution

Given information:

The value of trigonometric function $\tan \theta =\sqrt{3}$ .

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 $\tan \theta =\sqrt{3}$ .

Since, the tangent function is expressed as $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$ . Construct a right angle triangle with acute angle $\theta$ and whose adjacent side is 1 unit and opposite is $\sqrt{3}$ unit.

The figure obtained is provided below,

Observe that adjacent side is 1 unit and opposite is $\sqrt{3}$ unit.

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

  $\begin{array}{c}{1}^2+{\left(\sqrt{3}\right)}^2=x^2\\ x^2=4\\ x=2\end{array}$

Therefore, length of hypotenuse is $2$ 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,

  $\begin{array}{c}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ =\frac{\sqrt{3}}{2}\end{array}$

The value of cosine function is,

  $\begin{array}{c}cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ =\frac{1}{2}\end{array}$

The value of tangent function is,

  $\begin{array}{c}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\\ =\frac{\sqrt{3}}{1}\\ =\sqrt{3}\end{array}$

The value of cosecant function is,

  $\begin{array}{c}\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}}\\ =\frac{2}{\sqrt{3}}\\ =\frac{2\sqrt{3}}{3}\end{array}$

The value of secant function is,

  $\begin{array}{c}\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}\\ =\frac{2}{1}\\ =2\end{array}$

The value of cotangent function is,

  $\begin{array}{c}\cot \theta =\frac{\text{adjacent}}{\text{opposite}}\\ =\frac{1}{\sqrt{3}}\\ =\frac{\sqrt{3}}{3}\end{array}$

Hence, the value of trigonometric ratios are,

  $\sin \theta =\frac{\sqrt{3}}{2},cos\theta =\frac{1}{2},\tan \theta =\sqrt{3},\csc \theta =\frac{2\sqrt{3}}{3},\sec \theta =2\text{ and }\cot \theta =\frac{\sqrt{3}}{3}$ .