Chapter 6.3, Problem 46E

To determine

To calculate: The value of trigonometric ratios if $\cos \theta =-\frac{7}{12}$ and $\theta$ lies in Quadrant $\text{III}$ .

Answer to Problem 46E

The value of trigonometric ratios are,

  $\sin \theta =-\frac{\sqrt{95}}{12},cos\theta =-\frac{7}{12},\tan \theta =\frac{\sqrt{95}}{7},\csc \theta =-\frac{12\sqrt{95}}{95},\sec \theta =-\frac{12}{7}\text{ and }\cot \theta =\frac{7\sqrt{95}}{95}$ .

Explanation of Solution

Given information:

The value of trigonometric function $\cos \theta =-\frac{7}{12}$ and $\theta$ lies in Quadrant $\text{III}$ .

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 $\cos \theta =-\frac{7}{12}$ .

Since, the cosine function is expressed as $cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}}$ . Construct a right angle triangle with acute angle $\theta$ and whose adjacent side is 7 units and hypotenuse is 12 units.

The figure obtained is provided below,

Observe that adjacent side is 7 units and hypotenuse is 12 units.

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

  $\begin{array}{c}{x}^2+7^2={12}^2\\ x^2=144-49\\ x^2=95\\ x=\sqrt{95}\end{array}$

Therefore, length of opposite side is $\sqrt{95}$ 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 $\text{III}$ .

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.

Apply it, to estimate the value of trigonometric ratios,

The value of sine function is,

  $\begin{array}{c}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ =-\frac{\sqrt{95}}{12}\end{array}$

The value of cosine function is,

  $\begin{array}{c}cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ =-\frac{7}{12}\end{array}$

The value of tangent function is,

  $\begin{array}{c}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\\ =\frac{\sqrt{95}}{7}\end{array}$

The value of cosecant function is,

  $\begin{array}{c}\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}}\\ =-\frac{12}{\sqrt{95}}\\ =-\frac{12\sqrt{95}}{95}\end{array}$

The value of secant function is,

  $\begin{array}{c}\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}\\ =-\frac{12}{7}\end{array}$

The value of cotangent function is,

  $\begin{array}{c}\cot \theta =\frac{\text{adjacent}}{\text{opposite}}\\ =\frac{7}{\sqrt{95}}\\ =\frac{7\sqrt{95}}{95}\end{array}$

Hence, the value of trigonometric ratios are,

  $\sin \theta =-\frac{\sqrt{95}}{12},cos\theta =-\frac{7}{12},\tan \theta =\frac{\sqrt{95}}{7},\csc \theta =-\frac{12\sqrt{95}}{95},\sec \theta =-\frac{12}{7}\text{ and }\cot \theta =\frac{7\sqrt{95}}{95}$ .