Some trigonometric, exponential thing?

In summary, the conversation discusses the equivalence between the trigonometric form and exponential form of a function, specifically f(x)=A'sin(kx)+B'cos(kx) and f(x)=Ae^{ikx}+Be^{-ikx}. The conversation also mentions Euler's formula and the use of complex exponentials for finding complex solutions.
  • #1
M. next
382
0
How can we say:

f(x)=A'sin(kx)+B'cos(kx)

or equivalently

f(x)=Ae[itex]^{ikx}[/itex]+Be[itex]^{-ikx}[/itex]??

How are these two equivalent knowing that e[itex]^{ix}[/itex]=cosx+isinx

I don't get this?
 
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  • #2
Hi M. next! :smile:
M. next said:
How can we say:

f(x)=A'sin(kx)+B'cos(kx)

or equivalently

f(x)=Ae[itex]^{ikx}[/itex]+Be[itex]^{-ikx}[/itex]??

How are these two equivalent knowing that e[itex]^{ix}[/itex]=cosx+isinx

I don't get this?

They won't both be real.

Try Euler's formula

what do you get? :smile:
 
  • #3
it would be: A(coskx +isinkx)+B(coskx-isinkx)
which's (A+B)coskx+i(A-B)sinkx
.. A'coskx+iB'sinkx
where's did the "i" go?
 
  • #4
M. next said:
it would be: A(coskx +isinkx)+B(coskx-isinkx)
which's (A+B)coskx+i(A-B)sinkx

so B' = i(A-B) …

i told you they won't both be real! :biggrin:
 
  • #5
Sorry, i didn't check the site from since, I had some connection difficulties.
So, my final question, can this be done? Is the exponential form an alternative for the known trigonometric one?
And why do I use it? Why not keep it in trigonometric form. I am working on potential wells, free particles and so, if this information would help you answer my question.
 
  • #6
Hi M. next! :smile:
M. next said:
Is the exponential form an alternative for the known trigonometric one?
And why do I use it? Why not keep it in trigonometric form.

Yes, they're equally valid alternatives.

You use cos and sin, or real exponentials, if you're only interested in real solutions,

but you use complex exponentials if you're interested in complex solutions. :wink:
 
  • #7
Thanks, am grateful
 

What is trigonometry?

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles.

What are the main trigonometric functions?

The main trigonometric functions are sine, cosine, and tangent. These functions relate the angles of a right triangle to the lengths of its sides.

What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = ab^x, where a and b are constants and x is the variable. Exponential functions grow or decay at a constant rate.

How are trigonometric and exponential functions related?

Trigonometric functions can be used to model periodic phenomena, such as the growth and decay of exponential functions. Additionally, exponential functions can be used to solve certain trigonometric equations.

What are some real-life applications of trigonometry and exponential functions?

Trigonometry is used in fields such as engineering, physics, and astronomy to calculate distances, angles, and trajectories. Exponential functions are used in finance, population growth, and radioactive decay.

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