Chapter 6.2, Problem 21E

To determine

To calculate: The value of trigonometric ratios if $\cot \theta =1$ and also sketch the triangle with an acute angle $\theta$ .

Answer to Problem 21E

The value of trigonometric ratios are,

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

Explanation of Solution

Given information:

The value of trigonometric function $\cot \theta =1$ .

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 $\cot \theta =1$ .

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

The figure obtained is provided below,

Observe that adjacent side is 1 unit and opposite is 1 unit.

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

  $\begin{array}{c}{1}^2+1^2=x^2\\ x^2=2\\ x=\sqrt{2}\end{array}$

Therefore, length of hypotenuse is $\sqrt{2}$ 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,

  $\begin{array}{c}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ =\frac{1}{\sqrt{2}}\\ =\frac{\sqrt{2}}{2}\end{array}$

The value of cosine function is,

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

The value of tangent function is,

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

The value of cosecant function is,

  $\begin{array}{c}\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}}\\ =\frac{\sqrt{2}}{1}\\ =\sqrt{2}\end{array}$

The value of secant function is,

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

The value of cotangent function is,

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

Hence, the value of trigonometric ratios are,

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