Chapter 5.4, Problem 45E

To determine

To find: the exact value of the trigonometric expression when $\sin u=\frac{5}{13}$ and $\cos v=-\frac{3}{5}$ .

Answer to Problem 45E

  $\cos \left(u-v\right)=\frac{56}{65}$ .

Explanation of Solution

Given information:

Given expression,

  $\cos \left(u-v\right)$

Calculation:

Consider that $\sin u=\frac{5}{13}$ and $\cos v=-\frac{3}{5}$ , $u$ and $v$ are in quadrant II.

The sine of an angle is equal to opposite side over the hypotenuse. Since $\sin u=\frac{5}{13}$ and $u$ is quadrant II. So, the following figure will obtain:

  

Since cosine of an angle is equal to adjacent side over the hypotenuse and $u$ is in quadrant II.

Therefore:

  $\cos u=-\frac{12}{13}$

Since, $\cos v=-\frac{3}{5}$ and $v$ is in quadrant II. Consider the following figure:

  

Since the side of an angle is equal to opposite side over the hypotenuse. Therefore:

  $\sin v=\frac{4}{5}$

Apply the difference formula for cosine that is:

  $\cos \left(u-v\right)=\cos u\cos v+\sin u\sin v$

Here $u$ is interchanged by $v$ . Therefore

  $\cos \left(u-v\right)=\cos u\cos v+\sin u\sin v$

Substitute:

  $\sin u=\frac{5}{13},\cos u=-\frac{12}{13},\sin v=\frac{4}{5}$

And,

  $\cos v=-\frac{3}{5}$

So,

  $\begin{array}{c}\cos \left(u-v\right)=\left(\frac{-3}{5}\right)\left(\frac{-12}{13}\right)+\left(\frac{4}{5}\right)\left(\frac{5}{13}\right)\\ =\frac{36}{65}+\frac{20}{65}\\ =\frac{56}{65}\end{array}$

Therefore,

  $\cos \left(u-v\right)=\frac{56}{65}$ .