# User:345Kai

## Sine and Cosine

Fig. 1a - Sine and cosine of the angle θ in the unit circle of a cartesian coordinate system.

In a Cartesian coordinate system, consider the unit circle, which is of radius 1 and centered at the origin (see Figure 1a). The ray (blue) forming angle θ with the positive x-axis intersects the unit circle at a point whose x-coordinate (red) is the cosine and whose y-coordinate (green) is the sine of θ. This defines ${\displaystyle \sin \theta }$ and ${\displaystyle \cos \theta }$ for all angles between 0 and 360°. Sine and cosine of θ are real numbers between -1 and +1.

Fig. 1b - Angle θ in the second quadrant. The sine is positive, the cosine negative.
Fig. 1c - Angle θ in the third quadrant. Both sine and cosine are negative.
Fig. 1d - Angle θ in the fourth quadrant. The sine is negative, the cosine positive.

## Cartesian Coordinates

Fig 1 - Cartesian coordinate system with the points (5,12) marked in green, (-3,1) in red, (-1.5,-2.5) in blue and (0,0), the origin, in violet.

## other stuff

${\displaystyle {\begin{pmatrix}1&1\\-1&1\end{pmatrix}}}$

${\displaystyle {\begin{pmatrix}2&4\\-1&2\end{pmatrix}}}$

${\displaystyle {\begin{pmatrix}1&0&1\\0&1&0\end{pmatrix}}}$

${\displaystyle {\begin{pmatrix}0&0\\2&2\end{pmatrix}}}$

${\displaystyle {\begin{pmatrix}0\\1\\1\end{pmatrix}}}$

${\displaystyle {\vec {x}}'(t)=A\,{\vec {x}}(t)}$

${\displaystyle {\vec {x}}(t)=c_{1}e^{\lambda _{1}t}{\vec {v}}_{1}+\ldots +c_{n}e^{\lambda _{n}t}{\vec {v}}_{n}}$

${\displaystyle {\vec {x}}(k+1)=A\,{\vec {x}}(k)}$

${\displaystyle {\vec {x}}(k)=c_{1}\lambda _{1}^{n}{\vec {v}}_{1}+\ldots +c_{n}\lambda _{n}k{\vec {v}}_{n}}$

${\displaystyle \langle T{\vec {v}},{\vec {w}}\rangle =\langle {\vec {v}},T{\vec {w}}\rangle }$ Superscript text ${\displaystyle 3x+4x+20=}$ ${\displaystyle z^{m/n}=|z|^{m/n}e^{i\arg(z)m/n}}$