57 lines
1.4 KiB
Markdown
57 lines
1.4 KiB
Markdown
#Math #Trig
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# Euler's Formula
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Euler's formula states:
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$$
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e^{i \theta} = i\sin \theta + \cos \theta
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$$
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## Proof
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$$
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\frac d {d \theta} \frac {i \sin \theta + \cos \theta} {e^{i \theta}} \\
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= e^{-i\theta}(i \sin \theta + \cos \theta) \\
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= (e^{-i\theta})(i \sin \theta + \cos \theta)\prime + (e^{-i\theta}) \prime (i \sin \theta + \cos \theta) \\
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= (e^{-i\theta})(i \cos \theta - \sin \theta) - i(e^{-i\theta})(i \sin \theta + \cos \theta) \\
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= (e^{-i\theta})(i \cos \theta - \sin \theta) - (e^{-i\theta})(i \cos \theta - \sin \theta) \\
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= 0
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$$
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Therefore $\frac {i \sin \theta + \cos \theta} {e^{i \theta}}$ is a constant. Plug in $\theta = 0$, to get $\frac {i \sin \theta + \cos \theta} {e^{i \theta}} = 1$. Multiply both sides by $e^{i\theta}$ to get
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$$
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e^{i \theta} = i\sin \theta + \cos \theta
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$$
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## Euler's Identity
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Plug $\theta = π$ into Euler's Formula
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$$
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e^{i \pi} = i\sin \pi + \cos \pi \\
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e^{i \pi} = -1
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$$
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# Trig Functions Redefined
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Sine:
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$$
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e^{i \theta} = i\sin \theta + \cos \theta \\
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-e^{-i \theta} = -i\sin -\theta - \cos -\theta \\
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-e^{-i \theta} = i\sin \theta - \cos \theta \\
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e^{i\theta} - e^{-i\theta} = 2i \sin \theta \\
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\sin \theta = \frac {e^{i\theta} - e^{-i\theta}} {2i}
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$$
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Cosine:
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$$
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e^{i \theta} = i\sin \theta + \cos \theta \\
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e^{-i \theta} = i\sin -\theta + \cos -\theta \\
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e^{-i \theta} = -i\sin \theta + \cos \theta \\
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e^{i\theta} + e^{-i \theta} = 2\cos \theta \\
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\cos \theta = \frac {e^{i\theta} + e^{-i \theta}} 2
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$$ |