User:345Kai

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Sine and Cosine[edit]

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 \sin\theta and \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[edit]

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[edit]

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

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

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

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

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


\vec x ' (t) = A \, \vec x(t)

\vec x (t) = c_1 e^{\lambda_1 t}\vec v_1+\ldots+c_ne^{\lambda_n t} \vec v_n


\vec x (k+1) = A \, \vec x(k)

\vec x (k) = c_1  \lambda_1^n  \vec v_1+\ldots+c_n \lambda_n k \vec v_n

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