Properties of trigonometric functions.
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Inner product and norm |
We introduce the standard inner product and compute the distance and the angle between to vectors. |
By the trigonometric functions we understand the functions
sine (\(\sin\))
cosine (\(\cos\))
tangent (\(\tan\))
and their reciprocals.
Sine function: \(\sin :
\mathbb{R}\rightarrow\mathbb{R}\)
Properties:
antisymmetric, i.e. \(\sin(-x)=-\sin(x)\)
zeros: \(\{k\cdot\pi :
k\in\mathbb{Z}\}\)
periodic with period \(2\pi\),
i.e. \(\sin(x)=\sin(x+2\pi)\)
Cosine function \(\cos:
\mathbb{R}\rightarrow\mathbb{R}\)
Properties:
symmetric, i.e. \(\cos(x)=\cos(-x)\)
zeros: \(\{\frac{\pi}{2}+k\cdot\pi :
k\in\mathbb{Z}\}\)
periodic with period \(2\pi\),
i.e. \(\cos(x)=\cos(x+2\pi)\)
Tangent function \(\tan:
\{x\in\mathbb{R}:\cos(x)\neq 0\}\rightarrow\mathbb{R}\)
Properties:
Relations between trigonometric functions:
\(\cos(x)=\sin\left(x+\frac{\pi}{2}\right)\)
\(\sin\left(x\right)=\cos\left(x-\frac{\pi}{2}\right)\)
\(\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)\)
\(\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)\)
\(\sin^{2}(x)+\cos^{2}(x)=1\)
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