# Summation of sines and cosines questions

• mjordan2nd
In summary, the conversation discusses the relationship between two equations in a text that defines \sum E_{n} Cos \left( \frac{2 \pi X_{n}}{\lambda} \right) and \sum E_{n} Sin \left( \frac{2 \pi X_{n}}{\lambda} \right). The conversation concludes that it is fair to say \sum E_{n} Sin \left( \frac{2 \pi X_{n}}{\lambda} \right) = E Sin \phi based on Euler's identity. However, there may be constraints to this relationship, as not all combinations of E and \phi will work.
mjordan2nd
This is not a homework question per say, but rather a question I have about a text I am reading. In the text, they have defined

$$\sum E_{n} Cos \left( \frac{2 \pi X_{n}}{\lambda} \right) = E Cos \phi$$

If this is the case, is it fair to say that

$$\sum E_{n} Sin \left( \frac{2 \pi X_{n}}{\lambda} \right) = E Sin \phi$$

My text claims it is, though I can not figure out why. Any help would be appreciated.

Yes, it is fair to say, and the reason is Euler's identity,

$$e^{ix} = \cos(x) + i \sin (x)$$

Try starting with

$$\sum E_n \exp\left(\frac{i2 \pi X_n}{\lambda}\right)$$

and see what you come up with.

Ahh. Let me see if I can make an argument out of this.

$$\sum E_n \exp \left( \frac{i2 \pi X_n}{\lambda} \right) = \sum E_{n} Cos \left( \frac{2 \pi X_{n}}{\lambda} \right) + E_{n} i Sin \left( \frac{2 \pi X_{n}}{\lambda} \right)$$

But we can also say

$$\sum E_n \exp\left(\frac{i2 \pi X_n}{\lambda}\right) = Ee^{i \phi} = E cos \phi + i E sin \phi$$

By setting the real parts equal to each other and the imaginary parts equal to each other, we see that the definitions are consistent.

Right?

Looks good to me.

There have to be some contraints to this, as it doesn't hold in general. For example:

$$\cos(x)=\cos(-x)$$

but not

$$\sin(x)=\sin(-x)$$

Maybe I read more into the question than was actually there.

It is true that there is a solution $E, \phi$ that satisfies the two equations, and with the convention that E > 0, the solution is unique. (E = 0 is a degenerate exception, in which case $\phi$ can be anything.)

Reading the question more literally, it does not make sense to define $E, \phi$ by just the one equation

$$\sum E_{n} Cos \left( \frac{2 \pi X_{n}}{\lambda} \right) = E Cos \phi$$

as there are infinitely many combinations of $E, \phi$ that work.

Thanks for all the help, folks. Jbunniii your first interpretation was of the question was the one I intended.

## 1. What is the formula for finding the sum of sines and cosines?

The formula for finding the sum of sines and cosines is:
∑(sin(x) ± cos(x)) = sin(x) + sin(x + 1) + ... + sin(x + n) ± cos(x) + cos(x + 1) + ... + cos(x + n)
Where n is the number of terms to be added or subtracted.

## 2. How do I determine the period of a summation of sines and cosines?

The period of a summation of sines and cosines is equal to the least common multiple of the individual periods of the sine and cosine functions being added or subtracted.

## 3. Can the sum of sines and cosines be simplified?

Yes, the sum of sines and cosines can be simplified using trigonometric identities such as the double angle formula or the sum and difference formulas.

## 4. How do I solve a summation of sines and cosines with multiple angles?

To solve a summation of sines and cosines with multiple angles, use the appropriate trigonometric identities to simplify the expression and then apply the formula for finding the sum of sines and cosines.

## 5. What is the difference between a summation of sines and cosines and a Fourier series?

A summation of sines and cosines is a finite series of trigonometric functions, while a Fourier series is an infinite series that represents a periodic function using sines and cosines. Additionally, a Fourier series is used to represent a wider range of functions while a summation is typically used for simpler functions.

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