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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 $\endgroup$ 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. $\endgroup$ 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