bedastega4n3

2023-03-11

What is the difference between the graph of $f\left(x\right)=\mathrm{sin}\left(4x\right)$ and that of $f\left(x\right)=\mathrm{sin}\left(5x\right)\mathrm{cos}x-\mathrm{cos}\left(5x\right)\mathrm{sin}x$?

Nathanael Bates

Graph of $f\left(x\right)=\mathrm{sin}\left(4x\right)$ is as follows:
graph{sin(4x) [-5, 5, -2.46, 2.54]}
Graph of $g\left(x\right)=\mathrm{sin}\left(5x\right)\mathrm{cos}x-\mathrm{cos}\left(5x\right)\mathrm{sin}x$ is as follows:
graph{sin(5x)cosx-cos(5x)sinx [-5, 5, -2.46, 2.54]}
The similarity between the two is obvious. The reason is that $\mathrm{sin}\left(4x\right)=\mathrm{sin}\left(5x\right)\mathrm{cos}x-\mathrm{cos}\left(5x\right)\mathrm{sin}x$
This is as $\mathrm{sin}\left(A-B\right)=\mathrm{sin}A\mathrm{cos}B-\mathrm{cos}B\mathrm{sin}A$ and putting $A=5x$ and $B=x$ we get above result.

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