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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?

Narasimham
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Firgz
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  • Do you understand what I want to say or do I still need to clarify something? – Pedro Apr 07 '15 at 12:42

2 Answers2

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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 $20$.

enter image description here

Pedro
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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.

Narasimham
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