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

In summary, the conversation discusses solving a linear system differential equation using the exponential method and asks for the expansion of the exponential function. It also involves questions about the power series and identities of sine and cosine functions.
  • #1
Kenji Liew
25
0

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!
 
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  • #2
Kenji Liew said:

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?
[tex]\begin{align*}
\sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\
\cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
\end{align*}[/tex]

Or are you asking for the following identities?
[tex]\begin{align*}
\sin x &= \frac{e^{ix}-e^{-ix}}{2i} \\
\cos x &= \frac{e^{ix} + e^{-ix}}{2}
\end{align*}[/tex]
 
  • #3
Kenji Liew said:

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! + ...
 
  • #4
Also, you have posted what appears to be the same question twice, which is frowned upon in this and most other forums.
 

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

What is an exponential function?

An exponential function is a mathematical function that has the form f(x) = ab^x, where a and b are constants and x is the independent variable. The base, b, is usually a positive number greater than 1, and the function increases or decreases rapidly depending on the value of b.

What is the value of e^x?

The value of e^x is a mathematical constant equal to approximately 2.71828. It is the base of the natural logarithm and is commonly used in exponential functions and equations.

What is the value of e^(-1)?

The value of e^(-1) is equal to approximately 0.36788. This can be calculated by taking the reciprocal of e^1, which is equal to the value of e itself.

What is the relationship between exponential functions and sin x and cos x?

Exponential functions are often used to model periodic phenomena, such as the behavior of sin x and cos x. This is because exponential functions can oscillate between positive and negative values, just like sin x and cos x.

How can I solve an exponential function?

To solve an exponential function, you can use logarithms or take the natural logarithm of both sides of the equation. You can also use algebraic techniques, such as factoring and combining like terms, to isolate the variable and solve for its value.

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