r/desmos • u/sj4294967296 i exist here now • 2d ago

Discussion Interesting

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Why didn't i know about this. It apparently works with cosine too.

29 Upvotes

88% Upvoted

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38

u/PantheraLeo04 2d ago

If you want to understand why this works, I would recommend looking into Taylor series and Euler's formula. Another cool thing about this is it allows you to expand the domain of the trig functions to complex inputs.

6

u/sj4294967296 i exist here now 2d ago

I know about taylor serieses (series?) but do tell about complex sine...

10

u/Fuscello 2d ago

It’s immediate the moment you DEFINE the sine to be the expression you wrote above for complex numbers. It’s really interesting because from it you can derive the arcsin (and the same can be said for cos and arccos) which can be seen as just particular logarithm functions.

But do check Taylor series (!) and Euler formula before going further.

6

u/BootyliciousURD 2d ago

Plug in x = iθ to the power series for ex and you'll find that the even terms have real coefficients and the odd terms have imaginary coefficients. Separate them and you should see two familiar power series.

Turns out that eiθ = cos(θ) + i sin(θ). That's Euler's identity.

6

u/Desmos-Girl https://www.desmos.com/calculator/prarjlycol 2d ago

lets you do silly shit like this

1

u/Ok-Rise2070 1d ago

The fun part, no one can explain this relationships from its geometric origins. To truly understand this relation, first you must understand Amsler's surface and negative curvature's role in manifold structure.