An interesting (geometric) way of finding the derivatives of sin(x) and cos(x)

The method I use here was heavily inspired (just like most of Beyoncé's songs) by the one outlined in Tristan Needham's Visual Complex Analysis, where he used a similar way to find the derivative of

\tan{\theta}

.

So we start by constructing a right triangle of unit hypotenuse and angle

\theta

. Then the legs would be

\sin{\theta}

and

\cos{\theta}

. Then if we change the angle by

\Delta\theta

, keeping the hypotenuse constant of course, we get something similar to the attached diagram.

\Delta\sin{\theta}

and

\Delta\cos{\theta}

represent the change in

\sin{\theta}

and

\cos{\theta}

respectively, due to the change in the angle.

From the diagram, we can see that as

\Delta\theta

becomes infinitely small, the arc

\Delta\theta

approaches a straight line (dotted) and the radii (unit hypotenuses) become parallel. Not only that but also the arc/eventual straight line will become perpendicular to BOTH radii. So this small right triangle

\bigtriangleup DCC'

will eventually have hypotenuse

\Delta\theta

.

Angles

\theta

and

\angle ACD

are congruent as they are alternate angles. And since

\angle ACC'

is right,

m\angle CC'D = \theta

. So we can easily deduce the following:

\displaystyle \frac{\Delta\sin{\theta}}{\Delta \theta} \approx \cos{\theta} \implies \lim_{\Delta\theta\rightarrow 0} \frac{\Delta\sin{\theta}}{\Delta \theta} = \frac{d \sin{\theta}}{d \theta}=\cos{\theta}

, and

\displaystyle \frac{\Delta\cos{\theta}}{\Delta \theta} \approx -\sin{\theta} \implies \lim_{\Delta\theta\rightarrow 0} \frac{\Delta\cos{\theta}}{\Delta \theta} = \frac{d \cos{\theta}}{d \theta}= -\sin{\theta}

.
That negative sign comes from the fact that

\cos{\theta}

decreased here.
Done!

Hope you found it interesting!

(edited 8 years ago)

Reply 1

atsruser

11

Original post by gagafacea1

The method I use here was heavily inspired (just like most of Beyoncé's songs) by the one outlined in Tristan Needham's Visual Complex Analysis,

It's interesting to hear of a singer who draws on mathematics as the wellspring of her creativity. This is unusual and the only similar example with which I'm familiar was Elvis:

https://groups.google.com/forum/#!search/elvis$20physics/sci.physics/HrhBPS8_N_4/Ufe9q2SXZ_wJ

Reply 2

username1599987

OP

12

Original post by atsruser

It's interesting to hear of a singer who draws on mathematics as the wellspring of her creativity. This is unusual and the only similar example with which I'm familiar was Elvis:

https://groups.google.com/forum/#!search/elvis$20physics/sci.physics/HrhBPS8_N_4/Ufe9q2SXZ_wJ

lmaoooooo but actually meant that her songs are heavily inspired, not inspired by complex analysis. Though she does have "1+1" (song)!

Reply 3

Brit_Miller

14

Original post by gagafacea1

Tristan Needham's Visual Complex Analysis

This is an excellent book - it's my favourite on complex analysis.

Reply 4

username1599987

OP

12

Original post by Brit_Miller

This is an excellent book - it's my favourite on complex analysis.

FINALLY!

Somebody else who appreciates this book. I've always liked the idea of complex numbers, but this book just made me LOVE them. As a non-undergraduate, I found it really nice and easy to navigate without all the rigor that I don't have the time for.

An interesting (geometric) way of finding the derivatives of sin(x) and cos(x)

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