Chapter 5, Problem 77CR

To determine

To calculate: The exact value of the trigonometric expression.

Answer to Problem 77CR

The required exact value is $-\frac{4}{3}$ .

Explanation of Solution

Given information:

The trigonometric expression: $\tan \left(u-v\right)$

  $\sin u=\frac{4}{5}$

  $\cos v=-\frac{7}{25}$

Formula used:

Trigonometric formula:

  $\tan \left(u-v\right)=\frac{\tan u-\tan v}{1+\tan u\tan v}$

Pythagoras Theorem:

For a right-angle triangle;

  $\begin{array}{c}{\text{Hypotenuse}}^{\text{2}}={\text{Perpendicular}}^{\text{2}}+{\text{Base}}^{\text{2}}\\ h^2=p^2+b^2\end{array}$

Calculation:

The given trigonometric expression is $\tan \left(u-v\right)$ .

Also, $\sin u=\frac{4}{5}$

  $\cos v=-\frac{7}{25}$

To find the exact value of the given expression, use the formula $\tan \left(u-v\right)=\frac{\tan u-\tan v}{1+\tan u\tan v}$ .

Here, for $\sin u=\frac{4}{5}$ ;

Perpendicular, p = 4 and Hypotenuse, h = 5

By Pythagoras Theorem;

  $\begin{array}{c}{\text{Hypotenuse}}^{\text{2}}={\text{Perpendicular}}^{\text{2}}+{\text{Base}}^{\text{2}}\\ h^2=p^2+b^2\end{array}$

So, $\begin{array}{l}{h}^2=p^2+b^2\\ b=\sqrt{h^2-p^2}\\ b=\sqrt{5^2-4^2}\\ b=3\end{array}$

And, $\begin{array}{c}\cos u=\frac{b}{h}\\ =\frac{3}{5}\end{array}$

Also, for $\cos v=-\frac{7}{25}$ ;

Base, b = 4 and Hypotenuse, h = 5

By Pythagoras Theorem;

  $\begin{array}{c}{\text{Hypotenuse}}^{\text{2}}={\text{Perpendicular}}^{\text{2}}+{\text{Base}}^{\text{2}}\\ h^2=p^2+b^2\end{array}$

So, $\begin{array}{l}{h}^2=p^2+b^2\\ p=\sqrt{h^2-b^2}\\ p=\sqrt{{25}^2-{\left(-7\right)}^2}\\ b=24\end{array}$

And, $\begin{array}{c}\sin v=\frac{p}{h}\\ =\frac{24}{25}\end{array}$

Thus,

  $\begin{array}{c}\tan u=\frac{\sin u}{\cos u}\\ =\frac{\frac{4}{5}}{\frac{3}{5}}\\ =\frac{4}{\cancel{5}}\times \frac{\cancel{5}}{3}\\ =\frac{4}{3}\end{array}$

And,

  $\begin{array}{c}\tan v=\frac{\sin v}{\cos v}\\ =\frac{\frac{24}{25}}{-\frac{7}{25}}\\ =\frac{24}{\cancel{25}}\times \frac{\cancel{25}}{\left(-7\right)}\\ =-\frac{24}{7}\end{array}$

Now, substitute $\frac{4}{3}$ for $\tan u$ and $-\frac{24}{7}$ for $\tan v$ in the formula $\tan \left(u-v\right)=\frac{\tan u-\tan v}{1+\tan u\tan v}$ .

  $\begin{array}{c}\tan \left(u-v\right)=\frac{\left(\frac{4}{3}\right)-\left(-\frac{24}{7}\right)}{1+\left(\frac{4}{3}\right)\left(-\frac{24}{7}\right)}\\ =\frac{\frac{100}{21}}{\frac{-75}{21}}\\ =\frac{100}{\cancel{21}}\times \left(-\frac{\cancel{21}}{75}\right)\\ =-\frac{4}{3}\end{array}$

Hence, the required exact value is $-\frac{4}{3}$ .