Chapter 5.5, Problem 25E

To determine

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

Answer to Problem 25E

The exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $-\frac{\sqrt{3}}{2}$ , $-\frac{1}{2}$ and $\sqrt{3}$ respectively.

Explanation of Solution

Given information:

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

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 $\sec u=-2$ , Substitute the value $2$ for $\left(\text{hyp}\right)$ and $1$ for $\left(\text{adj}\right)$ in the Pythagoras theorem.

  $\begin{array}{c}\text{opp}=\sqrt{{\left(\text{hyp}\right)}^2-{\left(\text{adj}\right)}^2}\\ =\sqrt{{\left(2\right)}^2-{\left(1\right)}^2}\\ =\sqrt{4-1}\\ =\sqrt{3}\end{array}$ .

So, The value of $\text{opp}=\sqrt{3}$ .

Now substitute the values in the formula for sine

  $\begin{array}{c}\sin u=\frac{\text{opp}}{\text{hyp}}\\ =\frac{\sqrt{3}}{2}\end{array}$ .

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

Similarly substitute these values in the formula for tangent

  $\begin{array}{c}\tan u=-\frac{\text{opp}}{\text{adj}}\\ =-\frac{\sqrt{3}}{1}\\ =-\sqrt{3}\end{array}$ .

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

Now substitute both the values in the double angle formula

  $\begin{array}{c}\sin 2u=2\sin u\cos u\\ =2\times \frac{\sqrt{3}}{2}\times -\frac{1}{2}\\ =-\frac{\sqrt{3}}{2}\end{array}$ .

So, The exact value of $\sin 2u$ is $-\frac{\sqrt{3}}{2}$ .

Similarly, substitute the value in the formula

  $\begin{array}{c}\cos 2u=1-2{\sin }^{2}u\\ =1-2{\left(\frac{\sqrt{3}}{2}\right)}^2\\ =1-2\left(\frac{3}{4}\right)\\ =1-\frac{3}{2}\\ =-\frac{1}{2}\end{array}$ .

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

Similarly, substitute the value in the formula

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

  $\begin{array}{c}=\frac{-2\sqrt{3}}{-2}\\ =\sqrt{3}\end{array}$ .

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

Therefore, the exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $-\frac{\sqrt{3}}{2}$ , $-\frac{1}{2}$ and $\sqrt{3}$ respectively.