User:Timwi

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Lemma

My name is Timwi.

Proof

Self-evident.

$\blacksquare$

Corollary

I started the following pages:

Computer science

Trigonometry

Arccosine Logarithmic Formulation

$ \displaystyle \arccos x = -i \ln \left({ i \sqrt{1-x^2} + x }\right) $

Arcsine Logarithmic Formulation

$ \displaystyle \arcsin x = -i \ln \left({ \sqrt{1-x^2} + ix }\right) $

Arctangent Logarithmic Formulation

$ \displaystyle \arctan x = \frac 1 2 i \ln \left({ \frac{1-ix}{1+ix} }\right) $

Area of Triangle in Terms of Two Sides and Angle

$ \displaystyle \operatorname{Area}\left({ABC}\right) = \frac 1 2 a b \sin\theta $

Cosine Exponential Formulation

$ \displaystyle \cos x = \dfrac {e^{i x} + e^{-i x} } 2 $

Hyperbolic Cosine in terms of Cosine

$ \displaystyle \cos \left({ix}\right) = \cosh x $

Cosine of Sum/Proof using Exponential Formulation

$ \displaystyle \cos \left({a + b}\right) = \cos a \cos b - \sin a \sin b $

Hyperbolic Cosine Function is Even

$ \displaystyle \cosh \left({-x}\right) = \cosh x $

Hyperbolic Sine Function is Odd

$ \displaystyle \sinh \left({-x}\right) = -\sinh x $

Hyperbolic Tangent Function is Odd

$ \displaystyle \tanh \left({-x}\right) = -\tanh x $

Sine Exponential Formulation

$ \displaystyle \sin x = \frac 1 2 i \left({ e^{-i x} - e^{i x} }\right)$

Hyperbolic Sine in terms of Sine

$ \displaystyle \sin \left({ix}\right) = i \sinh x $

Sine of Sum/Proof using Exponential Formulation

$ \displaystyle \sin \left({a + b}\right) = \sin a \cos b + \cos a \sin b $

Tangent Exponential Formulation

$ \displaystyle \tan x = i \frac { 1 - e^{2ix} }{ 1 + e^{2ix} } $

Hyperbolic Tangent in terms of Tangent

$ \displaystyle \tan \left({ix}\right) = i \tanh x $