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 $\sin \theta$ 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

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

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

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

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

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

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

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

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

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

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

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

Jim Bryan Article

Jim Arthur Bryan (born December 6, 1951 in Belleville, Canada) is a Canadian mathematician working in the fields of homotopy theory, category theory, and number theory.

Biography

Jardine obtained his Ph.D. from the University of British Columbia in 1981, under the direction of Roy Douglas. Following a research fellowship at the University of Toronto and a Dickson instructorship at the University of Chicago, he joined the Department of Mathematics at the University of Western Ontario in 1984, where he is currently a professor.

From 2002 to 2016, Jardine held a Canada Research Chair in applied homotopy theory. Since 2008, he is fellow of the Fields Institute, and has been recognized with the Coxeter–James Prize in 1992 by the Canadian Mathematical Society.

Work

Jardine is known for his work on model category structures on simplicial presheaves.

External references

• Jardine's homepage at the University of Western Ontario
• 345Kai at the Mathematics Genealogy Project