Chapter 6.3, Problem 49E

To determine

To calculate: The value of trigonometric ratios if $\csc \theta =2$ and $\theta$ lies in Quadrant $\text{I}$ .

Answer to Problem 49E

The value of trigonometric ratios are,

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

Explanation of Solution

Given information:

The value of trigonometric function $\csc \theta =2$ and $\theta$ lies in Quadrant $\text{I}$ .

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 $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Coordinate plane is divided into four quadrants.

In the first quadrant all trigonometric functions that is $\sin \theta ,cos\theta ,\tan \theta ,\csc \theta ,\sec \theta \text{ and }\cot \theta$ are positive.

In the second quadrant only sine and cosecant trigonometric functions that is $\sin \theta \text{ and }\csc \theta$ are positive and remaining $cos\theta ,\tan \theta ,\sec \theta \text{ and }\cot \theta$ are negative.

In the third quadrant only tangent and cotangent trigonometric functions that is $\tan \theta \text{ and }\cot \theta$ are positive and remaining $cos\theta ,sin\theta ,\sec \theta \text{ and }\csc \theta$ are negative.

In the fourth quadrant only cosine and secant trigonometric functions that is $\cos \theta \text{ and }\sec \theta$ are positive and remaining $\sin \theta ,\tan \theta ,\csc \theta \text{ and }\cot \theta$ are negative.

Calculation:

Consider the provided value of trigonometric function $\csc \theta =2$ and $\theta$ lies in Quadrant $\text{I}$ .

Since, the cosecant function is expressed as $\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}}$ . Construct a right angle triangle with acute angle $\theta$ and whose opposite side is 1 unit and hypotenuse is 2 units.

The figure obtained is provided below,

Observe that opposite side is 1 unit and hypotenuse is 2 units.

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

  $\begin{array}{c}{x}^2+1^2=2^2\\ x^2=4-1\\ x^2=3\\ x=\sqrt{3}\end{array}$

Therefore, length of adjacent side is $\sqrt{3}$ 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 $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

It is provided that $\theta$ lies in Quadrant $\text{I}$ .

In the first quadrant all trigonometric functions that is $\sin \theta ,cos\theta ,\tan \theta ,\csc \theta ,\sec \theta \text{ and }\cot \theta$ are positive.

Apply it, to estimate the value of trigonometric ratios,

The value of sine function is,

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

The value of cosine function is,

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

The value of tangent function is,

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

The value of cosecant function is,

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

The value of secant function is,

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

The value of cotangent function is,

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

Hence, the value of trigonometric ratios are,

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