Chapter 5.5, Problem 24E

To determine

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

Answer to Problem 24E

The exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $-\frac{12}{37}$ , $\frac{35}{37}$ and $-\frac{12}{35}$ respectively.

Explanation of Solution

Given information:

The value of $\cot u=-6$ , $\frac{3\pi }{2}<u<2\pi$ .

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 $\cot u=-6$ , substitute the value $6$ for $\left(\text{adj}\right)$ and $1$ for $\left(\text{opp}\right)$ in the Pythagoras theorem.

  $\begin{array}{c}\text{hyp}=\sqrt{{\left(\text{opp}\right)}^2+{\left(\text{adj}\right)}^2}\\ =\sqrt{{\left(1\right)}^2+{\left(6\right)}^2}\\ =\sqrt{1+36}\\ =\sqrt{37}\end{array}$ .

So, The value of $\text{hyp}=\sqrt{37}$ .

Now substitute the values in the formula for sine

  $\begin{array}{c}\sin u=-\frac{\text{opp}}{\text{hyp}}\\ =-\frac{1}{\sqrt{37}}\\ =-\frac{\sqrt{37}}{37}\end{array}$ .

So, The exact value of $\sin u$ is $-\frac{\sqrt{37}}{37}$ .

Similarly substitute these values in the formula for cosine

  $\begin{array}{c}\cos u=\frac{\text{adj}}{\text{hyp}}\\ =\frac{6}{\sqrt{37}}\\ =\frac{6\sqrt{37}}{37}\end{array}$ .

So, the exact value of $\cos u$ is $\frac{6\sqrt{37}}{37}$ .

Now substitute both the values in the double angle formula

  $\begin{array}{c}\sin 2u=2\sin u\cos u\\ =2\times -\frac{\sqrt{37}}{37}\times \frac{6\sqrt{37}}{37}\\ =-\frac{12}{37}\end{array}$ .

So, The exact value of $\sin 2u$ is $-\frac{12}{37}$ .

Similarly, substitute the value in the formula

  $\begin{array}{c}\cos 2u=1-2{\sin }^{2}u\\ =1-2{\left(-\frac{\sqrt{37}}{37}\right)}^2\\ =1-2\left(\frac{1}{37}\right)\\ =1-\frac{2}{37}\\ =\frac{35}{37}\end{array}$ .

So, The exact value of $\cos 2u$ is $\frac{35}{37}$ .

Similarly, substitute the value in the formula

  $\begin{array}{c}\tan 2u=\frac{2\tan u}{1-{\tan }^{2}u}\\ =\frac{2\times -\frac{1}{6}}{1-{\left(-\frac{1}{6}\right)}^2}\\ =\frac{-\frac{1}{3}}{1-\frac{1}{36}}\end{array}$

  $\begin{array}{c}=-\frac{\frac{1}{3}}{\frac{35}{36}}\\ =-\frac{12}{35}\end{array}$ .

So, The exact value of $\tan 2u$ is $-\frac{12}{35}$ .

Therefore, the exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $-\frac{12}{37}$ , $\frac{35}{37}$ and $-\frac{12}{35}$ respectively.