Q13P - Mathematical Methods in Physical Sciences

Question

Verify each of the following by using equations (11.4), (12.2), and (12.3).

 $\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{3}\mathit{z}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{3}}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{-}\mathbf{3}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{z}\mathbf{\right)}$

Step-by-Step Solution

Answer

The equation $cos\left(3z\right)=4co{s}^3\left(z\right)-3cos\left(z\right)$ is verified using the equations (11.4), (12.2) and (12.3).

Step 1: Given Information

Given equation is $cos\left(3z\right)=4co{s}^3\left(z\right)-3cos\left(z\right)$ .

Step 2: Definition of Hyperbolic Function.

The term "Hyperbolic Function" refers to the relationship between a point on a hyperbola's distance from its origin and its coordinate axes, expressed as a function of an angle.

Step 3: Use exponential form to expand the equation

Given the equation is $cos\left(3z\right)=4co{s}^3\left(z\right)-3cos\left(z\right)$ .

The exponential form of the given equation is,

$cos\left(3z\right)=\frac{e^{\left(3zi\right)}+e^{\left(-3zi\right)}}{2}$                                                                                                 ….(1)

 

Let x(z),y(z)  be $e^{\left(zi\right)}and e^{\left(-zi\right)}$   respectively.

 

Substitute values in equation (1).

$$\begin{gathered} cos\left(3z\right)=\frac{x^3+y^3}{2} \\ =\frac{\left(x+y\right)\left(x^2-xy+y^2\right)}{2} \end{gathered}$$ 

 

Add $\left(2xy-2xy\right)$  in numerator.

 $$\begin{gathered} cos\left(3z\right)=\frac{\left(x+y\right)\left(x^2-xy+y^2+2xy-2xy\right)}{2} \\ =\frac{\left(x+y\right)\left(x^2-2xy+y^2-3xy\right)}{2} \\ =\frac{\left(x+y\right)\left[\left(x+y{)}^2-3xy\right)\right]}{2} \\ =\frac{{\left(x+y\right)}^3-3xy\left(x+y\right)}{2} \end{gathered}$$

 

Replace the value of  and .

 $$\begin{gathered} cos\left(3z\right)=\frac{{\left[exp\left(z\right)+exp\left(-z\right)\right]}^3}{2}-\frac{3exp\left(z\right)+exp\left(-z\right)\left[exp\left(z\right)+exp\left(-z\right)\right]}{2} \\ =\frac{4{\left[exp\left(z\right)+exp\left(-z\right)\right]}^3}{8}-\frac{3exp\left(z\right)+exp\left(-z\right)}{2} \\ =4{\left(\frac{exp\left(z\right)+exp\left(-z\right)}{2}\right)}^3-\frac{3exp\left(z\right)+exp\left(-z\right)}{2} \\ =4co{s}^3\left(z\right)-3cos\left(z\right) \end{gathered}$$

Hence, the equation is verified.