Chapter 7.3, Problem 31E

To determine

To find: the exact trigonometric value by using the given information.

Answer to Problem 31E

  $\frac{65}{56}$

Explanation of Solution

Given information:

  $\begin{array}{l}Trignometric\text{ function: }\sec \left(x-y\right)\\ \csc x=\frac{5}{3},0<x<\frac{\pi }{2}\\ \tan y=\frac{12}{5},0<y<\frac{\pi }{2}\end{array}$

Calculation:

Evaluate the value of $\sin x\text{ and }\cos x$ as shown below.

  $\begin{array}{c}\sin x=\frac{1}{\csc x}\left[\text{Reciprocal identity}\right]\\ =\frac{1}{\left(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)}\left[\text{Put }\csc x\right]\\ =\frac{3}{5}\\ \\ \cos x=\sqrt{1-{\sin }^{2}x}\left[\text{Pythagorean identity}\right]\\ =\sqrt{1-{\left(\frac{3}{5}\right)}^2}\left[\text{Put the value of }\sin x\right]\\ =\sqrt{\frac{16}{25}}\\ =\frac{4}{5}\end{array}$

Similarly evaluate the value of $\cos y\text{ and }\sin y$ as shown below.

  $\begin{array}{c}\sec y=\sqrt{1+{\tan }^{2}y}\left[\text{Pythagorean identity}\right]\\ \sec y=\sqrt{1+{\left(\frac{12}{5}\right)}^2}\left[\text{Put the value of }\tan y\right]\\ \sec y=\sqrt{\frac{169}{25}}\\ \frac{1}{\cos y}=\frac{13}{5}\left[\text{Reciprocal identity}\right]\\ \cos y=\frac{5}{13}\\ \\ \sin y=\sqrt{1-{\cos }^{2}y}\left[\text{Pythagorean identity}\right]\\ =\sqrt{1-{\left(\frac{5}{13}\right)}^2}\left[\text{Put the value of }\cos y\right]\\ =\sqrt{1-\frac{25}{169}}\\ =\sqrt{\frac{144}{169}}\\ =\frac{12}{13}\end{array}$

Now solve $\sec \left(x-y\right)$ as shown below.

  $\begin{array}{c}\cos \left(x-y\right)=\cos x\cos y+\sin x\sin y\left[\text{Difference identity for cosine}\right]\\ \cos \left(x-y\right)=\frac{4}{5}\cdot \frac{5}{13}+\frac{3}{5}\cdot \frac{12}{13}\left[\begin{array}{l}\text{Put the values for sine }\\ \text{and cosine functions}\end{array}\right]\\ \frac{1}{\sec \left(x-y\right)}=\frac{20+36}{65}\left[\text{Reciprocal identity: }\cos A=\frac{1}{\sec A}\right]\\ \frac{1}{\sec \left(x-y\right)}=\frac{56}{65}\\ \sec \left(x-y\right)=\frac{65}{56}\end{array}$

Thus the value of given trigonometric function is $\frac{65}{56}$ .