Q27P - Mathematical Methods in Physical Sciences

Question

Using the fact that a complex equation is really two real equations, find the double angle formulas $(for sin 2\theta , cos2\theta )$by using equation 10.2.

Step-by-Step Solution

Answer

The double angle formula for $\left(\sin 2\theta , \cos 2\theta \right)$ is $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$   

Step 1: Given Information

To find the double angle formulas for $\left(\sin 2\theta , \cos 2\theta \right)$ using equation 10.2.

Step 2: Definition of the complex number

Complex numbers are represented in terms of real numbers and imaginary numbers; a complex  can be written in the form of: 

$z=a+ib$  

 

Here a and b are real numbers, and i is the imaginary number which is known as iota, whose value is $\sqrt{-1}$.

Step 3: Finding an expression for and

Exponential form for z ;

$$\begin{gathered} U=e^{{\left(i\theta \right)}^2} \\ =e^{\left(2i\theta \right)} \\ =\cos 2\theta +i \sin 2\theta \end{gathered}$$ 

From the above, it can further have written as,

$$\begin{gathered} U=e^{{\left(i\theta \right)}^2} \\ ={\left[\cos \theta + \sin \theta \right]}^2 \end{gathered}$$ 

                                                                                   ……. (1)

 

 Simplifying the expression (1), we get,

  $U={\cos }^2 \left(\theta \right) -{\sin }^2 \left(\theta \right)+ 2 \sin \left(\theta \right) \cos \left(\theta \right) i$       ….… (2)

 

From (1) and (2) RE = RE and lm = lm  therefore;

$$\begin{gathered} \cos 2\theta ={\cos }^2\theta -{\sin }^2\theta \\ \sin 2\theta =2 \sin \theta \cos \theta \end{gathered}$$ 

 

 

Hence the formula will be, $\cos 2\theta ={\cos }^2 \left(\theta \right) -{\sin }^2 \left(\theta \right) , \sin \left(\theta \right) = 2 \sin \theta \cos \theta$.