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In the following example:

enter image description here

At the very last step, how does the author get that $\tan(\theta) = dy/dx$? To which $dy$ and $dx$ is this referring to? It can't be the same $dx$ that is labelled in the diagram, can it? Where do these $dy$ and $dx$ come from?

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    $\begingroup$ It is that dx. The dm is almost the hypothenuse, dy is then the vertical third leg of that triangle. tangent is a slope, by definition of tangent. Thus, there is an angle (very close to the vert opp angle of $\theta_1$) where it is the tangent of $\endgroup$
    – naturallyInconsistent
    Commented Apr 20, 2023 at 5:23
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    $\begingroup$ How is this homework? It's question about a derivation for equation of motion of a string. The close voters (and you know who you are) need to examine their views about helping other people understand physics as well as they do. $\endgroup$
    – John Rennie
    Commented Apr 20, 2023 at 7:59
  • $\begingroup$ In my experience the slope $dy/dx$ is usually denoted $\tan\psi$ to avoid conflation of $\psi$ with the polar angle $\theta$. $\endgroup$
    – J.G.
    Commented Apr 20, 2023 at 8:22

2 Answers 2

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By definition $\tan \theta = \frac yx $,

enter image description here

And if you look closely in given page it says :

On our diagram, however, the deformation has been exaggerated for clarity.

So what author has in mind is that deformation is infinitesimally small, so deformed arc is almost same by length as right triangle hypotenuse.

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tan theta =y/x

for a very small length tan theta =dy/dx

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