MULTIPLE ANGLE

A multiple angle is an angle which is written as a multiple of single function.

Example for multiple angles

\sin 3\theta ,\cos 2\theta

Multiple angle formulas:

1. \sin 2\theta  = 2\sin \theta \cos \theta

2. \cos 2\theta  = {\cos ^2}\theta  - {\sin ^2}\theta

\begin{array}{c}
 = 2{\cos ^2}\theta  - 1\\
 = 1 - 2{\sin ^2}\theta 
\end{array}

3. \tan 2\theta  = \frac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}

 

 

Question1 :

Solve2\cos x + \sin 2x = 0

Solution

Consider the equation

2\cos x + \sin 2x = 0

Use the multiple formulae as follow:

\sin 2\theta  = 2\sin \theta \cos \theta

Written as follows:

\begin{array}{c}
2\cos x + 2\sin x\cos x = 0\\
2\cos \left( {1 + \sin x} \right) = 0
\end{array}

Set the factors equal to zero as follows:

2\cos  = 0{\rm{ and }}1 + \sin x

Therefore, the solution is:

x = \frac{\pi }{2},\frac{{3\pi }}{2}{\rm{ and }}x = \frac{{3\pi }}{2}

The general solution for2\cos x + \sin 2x = 0is:

x = \frac{\pi }{2} + 2n\pi {\rm{ and }}x = \frac{{3\pi }}{2} + 2n\pi

Question 2:

Check the identity

{\cos ^2}\theta  = \frac{1}{2}\left[ {1 + \cos 2\theta } \right]

Solution

Use the multiple formulae as follows:

\begin{array}{c}
\cos 2\theta  = {\cos ^2}\theta  - {\sin ^2}\theta \\
 = 2{\cos ^2}\theta  - 1\\
2{\cos ^2}\theta  = 1 + \cos 2\theta \\
{\cos ^2}\theta  = \frac{1}{2}\left[ {1 + \cos 2\theta } \right]
\end{array}

Hence the identity is proved.

Post Author: E-Maths