Chapter 5.5, Problem 23E

To determine

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

Answer to Problem 23E

The exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $\frac{4}{5}$ , $\frac{3}{5}$ and $\frac{4}{3}$ respectively.

Explanation of Solution

Given information:

The value of $\tan u=\frac{1}{2}$ , $\pi <u<\frac{3\pi }{2}$ .

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

As $\tan u=\frac{1}{2}$ . Substitute the value $1$ for $\left(\text{opp}\right)$ and $2$ for $\left(\text{adj}\right)$ in the Pythagoras theorem

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{opp}\right)}^2+{\left(\text{adj}\right)}^2}\\ =\sqrt{{\left(1\right)}^2+{\left(2\right)}^2}\\ =\sqrt{1+4}\\ =\sqrt{5}\end{array}$ .

So, The value of $\text{hyp}=\sqrt{5}$ .

Now substitute the values in the formula for sine

  $\begin{array}{c}\sin u=-\frac{\text{opp}}{\text{hyp}}\\ =-\frac{1}{\sqrt{5}}\\ =-\frac{\sqrt{5}}{5}\end{array}$ .

So, The exact value of $\sin u$ is $-\frac{\sqrt{5}}{5}$ .

Similarly substitute these values in the formula for cosine

  $\begin{array}{c}\cos u=-\frac{\text{adj}}{\text{hyp}}\\ =-\frac{2}{\sqrt{5}}\\ =-\frac{2\sqrt{5}}{5}\end{array}$ .

So, the exact value of $\cos u$ is $-\frac{2\sqrt{5}}{5}$ .

Now substitute both the values in the double angle formula

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

So, The exact value of $\sin 2u$ is $\frac{4}{5}$ .

Similarly, substitute the value in the formula

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

Further simplify,

  $\begin{array}{c}=1-\frac{2}{5}\\ =\frac{3}{5}\end{array}$

So, The exact value of $\cos 2u$ is $\frac{3}{5}$ .

Similarly, substitute the value in the formula

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

  $\begin{array}{c}=\frac{1}{\frac{3}{4}}\\ =\frac{4}{3}\end{array}$ .

So, The exact value of $\tan 2u$ is $\frac{4}{3}$ .

Therefore, the exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $\frac{4}{5}$ , $\frac{3}{5}$ and $\frac{4}{3}$ respectively.