Chapter 5.5, Problem 21E

To determine

To calculate:The exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ using the double angle formulas.

Answer to Problem 21E

The exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $-\frac{24}{25}$ , $\frac{7}{25}$ and $-\frac{24}{7}$ respectively.

Explanation of Solution

Given information:

The value of $\sin u=-\frac{3}{5}$ , $\frac{3\pi }{2}<u<2\pi$ .

Formula used:

Use the double angle formulas for $\sin 2u$ , $\cos 2u$ and $\tan 2u$ as $2\sin u\cos u$ , $1-2{\sin }^{2}u$ and $\frac{2\tan u}{1-{\tan }^{2}u}$ respectively.

Calculation:

Given $\frac{3\pi }{2}<u<2\pi$ i.e. $u$ lies in fourth quadrant, therefore the value of $\sin u$ , $\tan u$ is negative and value of $\cos u$ is positive.

As $\sin u=-\frac{3}{5}$ . Substitute the value $5$ for $\left(\text{hyp}\right)$ and $3$ for $\left(\text{opp}\right)$ in the Pythagoras theorem.

  $\begin{array}{c}\text{adj}=\sqrt{{\left(\text{hyp}\right)}^2-{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(5\right)}^2-{\left(3\right)}^2}\\ =\sqrt{16}\\ =4\end{array}$ .

So, The value of $\text{adj}=4$ .

Now substitute the values in the formula for cosine

  $\begin{array}{c}\cos u=\frac{\text{adj}}{\text{hyp}}\\ =\frac{4}{5}\end{array}$ .

So, The exact value of $\cos u$ is $\frac{4}{5}$ .

Similarly substitute these values in the formula for tangent

  $\begin{array}{c}\tan u=\frac{\text{opp}}{\text{adj}}\\ =\frac{3}{4}\end{array}$ .

So, The exact value of $\tan u$ is $-\frac{3}{4}$ .

Now substitute both the values in the double angle formula

  $\begin{array}{c}\sin 2u=2\sin u\cos u\\ =2\times -\frac{3}{5}\times \frac{4}{5}\\ =-\frac{24}{25}\end{array}$ .

So, The exact value of $\sin 2u$ is $-\frac{24}{25}$ .

Similarly, substitute the value in the formula

  $\begin{array}{c}\cos 2u=1-2{\sin }^{2}u\\ =1-2{\left(-\frac{3}{5}\right)}^2\\ =1-2\left(\frac{9}{25}\right)\end{array}$ .

Further simplify,

  $\begin{array}{c}\cos 2u=1-\frac{18}{25}\\ =\frac{7}{25}\end{array}$

So, The exact value of $\cos 2u$ is $\frac{7}{25}$ .

Similarly, substitute the value in the formula

  $\begin{array}{c}\tan 2u=\frac{2\tan u}{1-{\tan }^{2}u}\\ =\frac{2\times -\frac{3}{4}}{1-{\left(-\frac{3}{4}\right)}^2}\\ =\frac{-\frac{3}{2}}{1-\frac{9}{16}}\end{array}$

  $\begin{array}{c}=\frac{-\frac{3}{2}}{\frac{7}{16}}\\ =-\frac{24}{7}\end{array}$ .

So, The exact value of $\tan 2u$ is $-\frac{24}{7}$ .

Therefore, the exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $-\frac{24}{25}$ , $\frac{7}{25}$ and $-\frac{24}{7}$ respectively.