Solve Exponential Function: e^x, e^(-1), Sin x, Cos x

[Kenji Liew]

Homework Statement

This topic is under linear system differential equation.Solve the system by using exponential method. Just want to ask the expansion of exponential function

Homework Equations

e^x=1+x+(x^2)/2!+(x^3)/3!+...

The Attempt at a Solution

then how about the e^(-1)=?
Besides what is the function of sin x and cos x in continued function (such in e^x)?
Thanks!

 

[n!kofeyn]

Homework Statement

The Attempt at a Solution

then how about the e^(-1)=?
Besides what is the function of sin x and cos x in continued function (such in e^x)?
Thanks!

I'm not really for sure what your question even is. What problem are you trying to solve? Are you asking for the power series for sine and cosine?
$$\begin{align*}<br /> \sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\<br /> \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots<br /> \end{align*}$$

Or are you asking for the following identities?
$$\begin{align*}<br /> \sin x &= \frac{e^{ix}-e^{-ix}}{2i} \\<br /> \cos x &= \frac{e^{ix} + e^{-ix}}{2}<br /> \end{align*}$$

 

[Mark44]

Homework Statement

This topic is under linear system differential equation.Solve the system by using exponential method. Just want to ask the expansion of exponential function

Homework Equations

e^x=1+x+(x^2)/2!+(x^3)/3!+...

The Attempt at a Solution

then how about the e^(-1)=?
Besides what is the function of sin x and cos x in continued function (such in e^x)?
Thanks!

e^-1 = 1 + (-1) + (-1)²/2! + (-1)³/3! + ... + (-1)ⁿ/n! + ...

 

[Mark44]

Also, you have posted what appears to be the same question twice, which is frowned upon in this and most other forums.