Chapter 5.5, Problem 26E

To determine

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

Answer to Problem 26E

The exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $\frac{4\sqrt{2}}{9}$ , $\frac{7}{9}$ and $\frac{4\sqrt{2}}{7}$ respectively.

Explanation of Solution

Given information:

The value of $\csc u=3$ , $0<u<\frac{ \pi }{2}$ .

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 $0<u<\frac{ \pi }{2}$ i.e. $u$ lies in first quadrant, therefore the value of all trigonometric identities is positive.

As $\csc u=3$ . Substitute the value $3$ for $\left(\text{hyp}\right)$ and $1$ for $\left(\text{opp}\right)$ in the Pythagoras theorem.

  $\begin{array}{c}\text{adj}=\sqrt{{\left(\text{hyp}\right)}^2-{\left(\text{opp}\right)}^2}\\ =\sqrt{{\left(3\right)}^2-{\left(1\right)}^2}\\ =\sqrt{9-1}\\ =2\sqrt{2}\end{array}$ .

So, The value of $\text{adj}=2\sqrt{2}$ .

Now substitute the values in the formula for cosine

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

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

Similarly substitute these values in the formula for tangent

  $\begin{array}{c}\tan u=\frac{\text{opp}}{\text{adj}}\\ =\frac{1}{2\sqrt{2}}\\ =\frac{\sqrt{2}}{4}\end{array}$ .

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

Now substitute both the values in the double angle formula

  $\begin{array}{c}\sin 2u=2\sin u\cos u\\ =2\times \frac{1}{3}\times \frac{2\sqrt{2}}{3}\\ =\frac{4\sqrt{2}}{9}\end{array}$ .

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

Similarly, substitute the value in the formula

  $\begin{array}{c}\cos 2u=1-2{\sin }^{2}u\\ =1-2{\left(\frac{1}{3}\right)}^2\\ =1-2\left(\frac{1}{9}\right)\\ =1-\frac{2}{9}\\ =\frac{7}{9}\end{array}$ .

So, The exact value of $\cos 2u$ is $\frac{7}{9}$ .

Similarly, substitute the value in the formula

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

  $\begin{array}{c}=\frac{\frac{\sqrt{2}}{2}}{\frac{14}{16}}\\ =\frac{4\sqrt{2}}{7}\end{array}$ .

So, the exact value of $\tan 2u$ is $\frac{4\sqrt{2}}{7}$ .

Therefore, the exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ are $\frac{4\sqrt{2}}{9}$ , $\frac{7}{9}$ and $\frac{4\sqrt{2}}{7}$ respectively.