Chapter 4, Problem 10CT

To determine

Find the exact values of the remaining five trigonometric functions of θ satisfying the given conditions.

Answer to Problem 10CT

The value of secant function for $\cos \theta =\frac{3}{5}$ is $\sec \theta =\frac{5}{3}$

Thevalue of sine function for $\cos \theta =\frac{3}{5}$ is, $\sin \theta =\frac{4}{5}$

Thevalue of the tangent function for $\cos \theta =\frac{3}{5}$ is $\tan \theta =\frac{4}{3}$

Thevalue of the cosecant function for $\cos \theta =\frac{3}{5}$ is $\csc \theta =\frac{5}{4}$

Thevalue of the co-tangent function for $\cos \theta =\frac{3}{5}$ is $\cot \theta =\frac{3}{4}$

Explanation of Solution

Given:

Consider the trigonometric function:

  $\cos \theta =\frac{3}{5}$

And the condition:

  $\tan \theta <0$

Use the reciprocal identity $\sec \theta =\frac{1}{\cos \theta },$ and hence,

  $\begin{array}{cc}\sec \theta & =\frac{1}{\frac{3}{5}}\\ & =\frac{5}{3}\end{array}$

Hence, the value of secant function for $\cos \theta =\frac{3}{5}$ is $\sec \theta =\frac{5}{3}$

Use the identity ${\sin }^2\theta =1-{\cos }^2\theta ,$ therefore,

  $\begin{array}{cc}\sin \theta & =\sqrt{1-{\cos }^2\theta }\\ & =\sqrt{1-{\left(\frac{3}{5}\right)}^2}\\ & =\sqrt{1-\frac{9}{25}}\\ & =\frac{4}{5}\end{array}$

Hence, the value of sine function for $\cos \theta =\frac{3}{5}$ is, $\sin \theta =\frac{4}{5}$

Use the quotient identity $\tan \theta =\frac{\sin \theta }{\cos \theta },$ therefore,

  $\begin{array}{cc}\tan \theta & =\frac{4/5}{3/5}\\ & =\frac{4}{5}\times \frac{5}{3}\\ & =\frac{4}{3}\end{array}$

Hence, the value of the tangent function for $\cos \theta =\frac{3}{5}$ is $\tan \theta =\frac{4}{3}$

Use the reciprocal identity $\csc \theta =\frac{1}{\sin \theta },$ therefore,

  $\begin{array}{cc}\csc \theta & =\frac{1}{4/5}\\ & =\frac{5}{4}\end{array}$

Hence, the value of the cosecant function for $\cos \theta =\frac{3}{5}$ is $\csc \theta =\frac{5}{4}$

Use the quotient identity $\cot \theta =\frac{\csc \theta }{\sec \theta },$ therefore

  $\begin{array}{cc}\cot \theta & =\frac{5/4}{5/3}\\ & =\frac{3}{4}\end{array}$

Hence, the value of the co-tangent function for $\cos \theta =\frac{3}{5}$ is $\cot \theta =\frac{3}{4}$