Chapter 5, Problem 23CT

To determine

Find the exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ .

Answer to Problem 23CT

  $\boxed{\sin 2u=\frac{-20}{29}}$

  $\boxed{\cos 2u=\frac{-21}{29}}$

  $\boxed{\tan 2u=\frac{21}{29}}$

Explanation of Solution

Given information:

Use the figure to find the exact values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ .

Calculation:

Let us consider the following figure,

  

Since, the point $\left(2,-5\right)$ lies in the fourth quadrant, the value of angle $u$ can be written as $u={\tan }^{-1}m$

Where $m$ is the slope of the line joining the point $\left(2,-5\right)$ and $\left(0,0\right)$ .

The slope of the line joining the point $\left(2,-5\right)$ and $\left(0,0\right)$ is calculated as,

  $m=\frac{-5-0}{\begin{array}{l}2-0\\ =\frac{-5}{2}\end{array}}$

Since $u={\tan }^{-1}m$ , then the value of the trigonometric functions can be calculated from the use of trigonometric identities.

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

  $=\frac{2}{\sqrt{29}}$

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

  $=\frac{-5}{\sqrt{29}}$

As the values of the required trigonometric ratios is known to calculate the values of $\sin 2u$ , $\cos 2u$ and $\tan 2u$ ,we will now calculate the required results.

  $\begin{array}{l}\sin 2u=2\sin u\cos u\\ \sin 2u=2\left(\frac{-5}{\sqrt{29}}\right)\left(\frac{2}{\sqrt{29}}\right)\\ \boxed{\sin 2u=\frac{-20}{29}}\\ \cos 2u={\cos }^{2}u-{\sin }^{2}u\\ \cos 2u={\left(\frac{2}{\sqrt{29}}\right)}^2-{\left(\frac{-5}{\sqrt{29}}\right)}^2\\ \cos 2u=\frac{4}{29}-\frac{25}{29}\\ \boxed{\cos 2u=\frac{-21}{29}}\\ \tan 2u=\frac{\sin 2u}{\cos 2u}\end{array}$

  $\begin{array}{l}\tan 2u=\frac{\left(\frac{-20}{29}\right)}{\left(\frac{-21}{29}\right)}\\ \boxed{\tan 2u=\frac{21}{20}}\end{array}$

Hence, $\boxed{\sin 2u=\frac{-20}{29}}$ , $\boxed{\cos 2u=\frac{-21}{29}}$ and $\boxed{\tan 2u=\frac{21}{29}}$ .