Euler's Formula |
\(e^{i\theta} = \cos(\theta)+i\sin(\theta)\) |
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Euler's Formula is where exponentials, complex numbers and trig all combine into one elegant formula, \(e^{i\theta} = \cos(\theta)+i\sin(\theta)\). This may not seem very useful but in electronics, for example, we convert trig functions to complex exponental form, using this equation, to make calculations easier. But how do we do that?
The idea is replace \(\theta\) with \(-\theta\) to get \(e^{-i\theta} = \cos(-\theta)+i\sin(-\theta) = \cos(\theta)-i\sin(\theta)\). Then we add this to \(e^{i\theta}\) and solve for cosine. Similarly, we subtract them and solve for sine. This process yields these equations.
\(\displaystyle{ \cos(\theta) = \frac{e^{i\theta}+e^{-i\theta}}{2} }\) |
\(\displaystyle{ \sin(\theta) = \frac{e^{i\theta}-e^{-i\theta}}{2i} }\) |
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Rather than just taking Euler's Formula for granted, let's look closely at what it is telling us. Here is a great video giving an intuitive explanation of the equation and what it means. This guy is great. He takes complicated concepts and makes them easier to see and understand.
video by BetterExplained |
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Now all this may seem a bit overwhelming but let's watch a great video explaining more intuitively why \(e^{i\pi}=-1\) holds.
video by 3Blue1Brown |
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Proofs of Euler's Formula
These proofs require calculus.
video by Derek Owens |
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video by Dave Academy |
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video by Dr Chris Tisdell |
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Here is a playlist of the videos on this page.
Really UNDERSTAND Precalculus
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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