Simplify the following equation [Complex Numbers]

In summary, the conversation is about Laplace Transforms in differential equations and a homework problem involving simplifying a series of cosine terms over complex numbers. The participants discuss using the real part of exp(i*n*theta) and relating the problem to a geometric series to find the solution.
  • #1
jcurl
2
0

Homework Statement


I'm in differential equations right now and we are about to start Laplace Transforms. Our homework is over complex numbers:

Simplify the following equation:
[itex] 1+cos(\theta)+cos(2\theta)+cos(3\theta)+...+cos(n\theta) [/itex]


Homework Equations





The Attempt at a Solution


I have no idea where to start. My only guess would be to do this: [itex] \sum_{i=0}^n cos(n\theta) [/itex] but I feel like that's way to easy and not what he is asking for.
 
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  • #2
If you are going to start Laplace transforms, you likely know about complex numbers. cos(n*theta) is the real part of exp(i*n*theta), yes? Can you relate your question to a geometric series?
 
  • #3
Dick said:
If you are going to start Laplace transforms, you likely know about complex numbers. cos(n*theta) is the real part of exp(i*n*theta), yes? Can you relate your question to a geometric series?

That's what I was thinking, but I'm not sure how to only get the [itex] cos [/itex] value. Since [itex] e^{i\theta} = cos(\theta) + isin(\theta) [/itex]
 
  • #4
Hint: scroll up and read Dick's post carefully.
 

What is a complex number?

A complex number is a number that is expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, equal to the square root of -1.

How do you simplify a complex number?

To simplify a complex number, you need to combine like terms, just as you would with a regular algebraic expression. This means combining the real numbers (a) and the imaginary numbers (bi).

What is the difference between a real and an imaginary number?

A real number is any number that can be represented on a number line, including both positive and negative numbers. An imaginary number, on the other hand, is any number that includes the imaginary unit i. It cannot be represented on a number line and is often used to represent complex numbers.

How do you add or subtract complex numbers?

To add or subtract complex numbers, you simply combine the real parts and the imaginary parts separately. For example, (3 + 2i) + (1 + 5i) = (3 + 1) + (2i + 5i) = 4 + 7i.

Can you multiply or divide complex numbers?

Yes, you can multiply or divide complex numbers just like you would with regular algebraic expressions. To multiply, you use the FOIL method, and to divide, you multiply by the complex conjugate of the denominator. For example, (3 + 2i) x (1 + 5i) = 3 + 15i + 2i + 10i^2 = (3 - 10) + (15 + 2)i = -7 + 17i.

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