Chapter 5.5, Problem 22E

To determine

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

Answer to Problem 22E

The exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $-\frac{4\sqrt{5}}{9}$ , $-\frac{1}{9}$ and $4\sqrt{5}$ respectively.

Explanation of Solution

Given information:

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

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

As $\cos u=-\frac{2}{3}$ ,substitute the value $2$ for $\left(\text{adj}\right)$ and $3$ for $\left(\text{hyp}\right)$ in the Pythagoras theorem

  $\begin{array}{c}\text{opp}=\sqrt{{\left(\text{hyp}\right)}^2-{\left(\text{adj}\right)}^2}\\ =\sqrt{{\left(3\right)}^2-{\left(2\right)}^2}\\ =\sqrt{9-4}\\ =\sqrt{5}\end{array}$ .

So, the value of $\text{opp}=\sqrt{5}$ .

Now substitute the values in the formula for sine

  $\begin{array}{c}\sin u=\frac{\text{opp}}{\text{hyp}}\\ =\frac{\sqrt{5}}{3}\end{array}$ .

So, the exact value of $\sin u$ is $\frac{\sqrt{5}}{3}$ .

Similarly substitute these values in the formula tangent

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

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

Now substitute both the values in the double angle formula

  $\begin{array}{c}\sin 2u=2\sin u\cos u\\ =2\times \frac{\sqrt{5}}{3}\times -\frac{2}{3}\\ =-\frac{4\sqrt{5}}{9}\end{array}$ .

So, the exact value of $\sin 2u$ is $-\frac{4\sqrt{5}}{9}$ .

Similarly, substitute the value in the formula

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

Further simplify,

  $\begin{array}{c}=1-\frac{10}{9}\\ =-\frac{1}{9}\end{array}$

So, the exact value of $\cos 2u$ is $-\frac{1}{9}$ .

Similarly, substitute the value in the formula

  $\begin{array}{c}\tan 2u=\frac{2\tan u}{1-{\tan }^{2}u}\\ =\frac{2\times -\frac{\sqrt{5}}{2}}{1-{\left(-\frac{\sqrt{5}}{2}\right)}^2}\\ =\frac{-\sqrt{5}}{1-\frac{5}{4}}\end{array}$

  $\begin{array}{c}=\frac{-\sqrt{5}}{-\frac{1}{4}}\\ =4\sqrt{5}\end{array}$ .

So, the exact value of $\tan 2u$ is $4\sqrt{5}$ .

Therefore, the exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $-\frac{4\sqrt{5}}{9}$ , $-\frac{1}{9}$ and $4\sqrt{5}$ respectively.