I currently have increments of 0.1 from 1 to 20 for x values. I have produced graphs for sin and cos but I am now looking into the effects of multiplying them with numbers in front. Would anyone be able to explain the effects of multiplying $cos$ and $sin$ with a number using the example below?
$$ 1 \cos(x) + 2 \;\sin(3x) + 4 \;\cos(5x) $$
I understand for example how I would plot $ \cos(3x) $ but I am unsure how to plot
$$ 4\; \cos(5x)$$
What does the 4 before \cos mean?
On
Sure, these numbers which you want to multiply with are called coefficients.
The waves you already sketched have half heights $1$ up and below x-axis. By multiplying with $2$ the height (also called amplitude) doubles.With $4$ it grows uniformly four times everywhere.
However if you change the number inside the bracket ( call it argument) the entire wave gets compressed or elongated. So the effect of argument multiple is to expand or squeeze the curve along x-axis.
After understanding these two effects separately then only go for adding them.
A single wave has equation
$$ y = A \sin ( \frac{2 \pi x}{ \lambda} ) $$
where $\lambda $ is the wave length of the periodic wave.
It is the amplitude, it will make the upper & lower bounds of the function on the $y$-axis higher and lower. Normally $\cos x$ and $\sin x$ are between $-1$ and $1$ on the $y$-axis. By taking $4 \cos x$ and $4 \sin x$, your values will be between $-4$ and $4$. You could say that it "streches out" your function along the $y$-axis. Everything gets out of proportion and gets a bit longer in the direction of the $y$-axis.
In the figure below, a high amplitude is an amplitude of $100$, a medium amplitude is an amplitude of $60$ and a low amplitude is an amplitude of $20$.