Chapter 6.3, Problem 73E

To determine

To calculate: The value of trigonometric ratios of the angle $\theta$ provided in the figure.

  

Answer to Problem 73E

The value of trigonometric ratios are,

  $cos\theta =\left|OP\right|,\tan \theta =\left|QR\right|,\csc \theta =\left|OS\right|,\sec \theta =\left|OQ\right|\text{and }\cot \theta =\left|SR\right|$ .

Explanation of Solution

Given information:

The radius of the circle is 1 with center O , segment QS is tangent to the circle at R , and right angled at $\angle SOQ$ .

  

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 radius of the circle is 1 with center O , segment QS is tangent to the circle at R , and right angled at $\angle SOQ$ .

  

It is provided that $\angle SOQ=90°$ and $\angle ROQ=\theta$ then $\angle SOR=90°-\theta$ .

Also QS is tangent at R, so $\angle SRO=\angle QRO=90°$ , therefore, $\angle OSR=\theta$ .

Recall that the trigonometric ratios for a right angle triangle are defined as,

  $cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}},\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}},\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}$ and $\cot \theta =\frac{\text{adjacent}}{\text{opposite}}$ .

Apply it, to estimate the value of trigonometric ratios, for $△OSR$ ,

The value of cosecant function is,

  $\begin{array}{c}\csc \theta =\frac{\text{hypotenuse}}{\text{opposite}}\\ =\frac{\left|OS\right|}{\left|OR\right|}\\ =\frac{\left|OS\right|}{1}\\ =\left|OS\right|\end{array}$

The value of cotangent function is,

  $\begin{array}{c}\cot \theta =\frac{\text{adjacent}}{\text{opposite}}\\ =\frac{\left|SR\right|}{\left|OR\right|}\\ =\frac{\left|SR\right|}{1}\\ =\left|SR\right|\end{array}$

For $△OPR$ ,

The value of cosine function is,

  $\begin{array}{c}cos\theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ =\frac{\left|OP\right|}{\left|OR\right|}\\ =\frac{\left|OP\right|}{1}\\ =\left|OP\right|\end{array}$

For $△OQR$ ,

The value of tangent function is,

  $\begin{array}{c}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\\ =\frac{\left|QR\right|}{\left|OR\right|}\\ =\frac{\left|QR\right|}{1}\\ =\left|QR\right|\end{array}$

The value of secant function is,

  $\begin{array}{c}\sec \theta =\frac{\text{hypotenuse}}{\text{adjacent}}\\ =\frac{\left|OQ\right|}{\left|OR\right|}\\ =\frac{\left|OQ\right|}{1}\\ =\left|OQ\right|\end{array}$

Hence, the value of trigonometric ratios are,

  $cos\theta =\left|OP\right|,\tan \theta =\left|QR\right|,\csc \theta =\left|OS\right|,\sec \theta =\left|OQ\right|\text{and }\cot \theta =\left|SR\right|$ .