Q7P - Mathematical Methods in Physical Sciences

Question

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

sinh 2z=2 sinh z cosh z

Step-by-Step Solution

Answer

The equation sinh 2z=2 sinh z cosh z is verified using the equations (11.4), (12.2) and (12.3).

Step 1: Given information

The given equation is sinh 2z=2 sinh z cosh z.

Step 2: Definition of Hyperbolic Function.

A relationship between the distances from a point on a hyperbola to the origin and the coordinate axes, represented as a function of an angle is called Hyperbolic Function.

Step 3: Use exponential form to expand the equation

The exponential form of the given equation is,

 $\sin h 2z=\frac{e^{2z}-e^{-2z}}{2}$                                                                                              …. (1)

 

Use identity $x^2-y^2=\left(x+y\right)\left(x-y\right)$ to split the numerator of equation (1).

$\sinh 2\frac{\left(e^z+e^{-z}\right)\left(e^z-e^{-z}\right)}{2}$                                                                                       …(2)

 

Multiply the equation (2) by $\frac{2}{2}$.

$$\begin{gathered} \sin h 2z=\frac{\left(e^z+e^{-z}\right)\left(e^z-e^{-z}\right)}{2}\times \frac{2}{2} \\ \sin h 2z=2\times \frac{\left(e^z+e^{-z}\right)}{2}\times \frac{\left(e^z-e^{-z}\right)}{2} \end{gathered}$$

Put $\frac{\left(e^z-e^{-z}\right)}{2}=\cos h z$.

Put $\frac{\left(e^z-e^{-z}\right)}{2}=s\mathrm{inh} \mathrm{z}$.

sinh z=2 cosh z. sinh z

Hence the equation is verified.