Sin(x) + Sin(y)

Question

When you add sound waves you are basically adding sine and cosine of certain multiples of x.

Is Sin(x) + Sin(y) ... + Sin(n) = Sin(x+y...+n)? Is the same true for summation of cosines?

I am making a game where you can help the emperors of the solar systems add the voices of all the numbers(and when you get to science organisms like mice and trees, atoms and molecules, and physical units) on a particular planet subtracting out any baby cries and assign that as the voice of the planet.

This requires summation of sines and cosines.

Also can you take the Sin or Cos of a matrix or vector? If so would that be equal to the Sin or Cos of the individual components of the matrix or vector?

Answer 1

No... For one thing $\sin(x)$ is bounded, it can be at most 1 and must be at least -1. So, for example

$$\sin(\pi/2)+\sin(\pi/2) = 2$$

but there is no $t$ which satisfies $\sin(t)=2$.

Answer 2

No, in general $\sin(x+y) \ne \sin(x) + \sin(y)$. (Take $x = y = \pi/4$ or $45$ degrees as an example).

The rules governing addition of sines and cosines are called "the addition rules for trigonometric functions". You can find a million discussions of them with Google.

Answer 3

A function satisfying $f(x+y)=f(x)+f(y)$ for any real numbers needs to be a linear function (of the type $f(x)=a\cdot x$), which the sine obviously is not. Other Answers have given simple counter examples.

Your second question is a lot harder to answer. And in fact: yes, there is a well defined way of taking the sine of a Matrix. One way to define this is the fact that $$\sin(x)=\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}$$ using this it is not hard to generalize to matrices (at least some matrices) just by replacing the $x$ by a Matrix. Be however aware, that is definition is not the sine applied to the components!

I am not aware of a proper way of defining a sine for vectors as there is no notion of multiplying more than two vectors. A component-wise definition would be possible but I doubt it would have nice properties.

Answer 4

In general, a sum of sines over a set of values will not equal the sine of the sum of the values:

$$\exists x,y\in\mathbb{R}:\sin{(x)}+\sin{(y)}\neq\sin{(x+y)}.$$

That said, there are situations where you can get away with using the equation as an approximation. For small values of $x$,

$$\sin{x}\approx x.$$

Thus, if all the terms are sufficiently small, you'll get

$$\sin{(x)}+\sin{(y)}\approx x+y\approx \sin{(x+y)},$$

similarly for sums of three terms, etc. Of course, without knowing the context in which you were wanting to use this formula I can't say one way or the other if this approximation is an option. You'll have to judge for yourself.