Chapter A6, Problem 7E

To determine

To simplify:the given equation.

Answer to Problem 7E

Theexponential form is $e^{5x}.$

Explanation of Solution

Given:

  $\text{sinh 5}x+\text{cosh 5}x.$

Calculation:

It is known that,

  $\text{sinh }x=\frac{e^x-e^{-x}}{2}$ and $\text{cosh }x=\frac{e^x+e^{-x}}{2}.$

Therefore,

  $\text{sinh 5}x+\text{cosh 5}x=\frac{e^{5x}-e^{-5x}}{2}+\frac{e^{5x}+e^{-5x}}{2}.$

  = $e^{5x}.$

This can also be verified graphically by plotting a graph of

  $y=\text{sinh 5}x+\text{cosh 5}x-e^{5x}.$

Since equal terms are subtracted from each other which should result in zero. Therefore, If

This graph comes similar to $y=0$ , then our solution is correct.

The graph of the function $y=\sinh \text{ 5}x+\cosh \text{ 5}x$ is shown below.

  

The graph of the function $y=e^x$ is shown below.

  

The graph of the function $y=\text{sinh 5}x+\text{cosh 5}x-e^{5x}$ is shown below.

  

Conclusion:

The exponential form is $e^{5x}$ .