Monday, July 30, 2018

calculus - Different ways finding the derivative of $sin$ and $cos$.



I am looking for different ways of differentiating $\sin$ and $\cos$, especially when using the geometric definition, but ways that use other defintions are also welcome.



Please include the definition you are using.




I think there are several ways to do this I didn't find, that make clever use of trigonometric identities or derivative facts. I am looking for these for several reasons. Firstly, it is just interesting to see. Secondly, there may be interesting techniques that can also be applied in a clever way to other derivative problems. It is also interesting to see how proofs can come form completely different fields of mathematics or even from physics.



I have included several solutions in the answers.


Answer



Consider the image from this "proof without words",



Derivative of $\sin(x)$



Asymptotically, the angle between the black radius and the red vertical line is complementary to both angles marked as $\theta$. Thus, asymptotically, those angles are equal, and the two red triangles are similar. Therefore, by similar triangles,

$$
\frac{\mathrm{d}\sin(\theta)}{\mathrm{d}\theta}=\frac{\cos(\theta)}{1}
$$
To get the derivative of $\cos(\theta)$, recall that $\cos(\theta)=\sin\left(\frac\pi2-\theta\right)$ and $\sin(\theta)=\cos\left(\frac\pi2-\theta\right)$. Then the Chain Rule says
$$
\begin{align}
\frac{\mathrm{d}}{\mathrm{d}\theta}\cos(\theta)
&=\frac{\mathrm{d}}{\mathrm{d}\theta}\sin\left(\frac\pi2-\theta\right)\\
&=-\cos\left(\frac\pi2-\theta\right)\\[3pt]
&=-\sin(\theta)

\end{align}
$$


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