Fig. 1a - Sine and cosine of the angle θ しーた 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 しーた 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 しーた . This defines
sin
θ しーた
{\displaystyle \sin \theta }
and
cos
θ しーた
{\displaystyle \cos \theta }
for all angles between 0 and 360°. Sine and cosine of θ しーた are real numbers between -1 and +1.
Fig. 1b - Angle θ しーた in the second quadrant. The sine is positive, the cosine negative.
Fig. 1c - Angle θ しーた in the third quadrant. Both sine and cosine are negative.
Fig. 1d - Angle θ しーた 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.
(
1
1
−
1
1
)
{\displaystyle {\begin{pmatrix}1&1\\-1&1\end{pmatrix}}}
(
2
4
−
1
2
)
{\displaystyle {\begin{pmatrix}2&4\\-1&2\end{pmatrix}}}
(
1
0
1
0
1
0
)
{\displaystyle {\begin{pmatrix}1&0&1\\0&1&0\end{pmatrix}}}
(
0
0
2
2
)
{\displaystyle {\begin{pmatrix}0&0\\2&2\end{pmatrix}}}
(
0
1
1
)
{\displaystyle {\begin{pmatrix}0\\1\\1\end{pmatrix}}}
x
→
′
(
t
)
=
A
x
→
(
t
)
{\displaystyle {\vec {x}}'(t)=A\,{\vec {x}}(t)}
x
→
(
t
)
=
c
1
e
λ らむだ
1
t
v
→
1
+
…
+
c
n
e
λ らむだ
n
t
v
→
n
{\displaystyle {\vec {x}}(t)=c_{1}e^{\lambda _{1}t}{\vec {v}}_{1}+\ldots +c_{n}e^{\lambda _{n}t}{\vec {v}}_{n}}
x
→
(
k
+
1
)
=
A
x
→
(
k
)
{\displaystyle {\vec {x}}(k+1)=A\,{\vec {x}}(k)}
x
→
(
k
)
=
c
1
λ らむだ
1
n
v
→
1
+
…
+
c
n
λ らむだ
n
k
v
→
n
{\displaystyle {\vec {x}}(k)=c_{1}\lambda _{1}^{n}{\vec {v}}_{1}+\ldots +c_{n}\lambda _{n}k{\vec {v}}_{n}}
⟨
T
v
→
,
w
→
⟩
=
⟨
v
→
,
T
w
→
⟩
{\displaystyle \langle T{\vec {v}},{\vec {w}}\rangle =\langle {\vec {v}},T{\vec {w}}\rangle }
Superscript text
3
x
+
4
x
+
20
=
{\displaystyle 3x+4x+20=}
z
m
/
n
=
|
z
|
m
/
n
e
i
arg
(
z
)
m
/
n
{\displaystyle z^{m/n}=|z|^{m/n}e^{i\arg(z)m/n}}
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 .
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.[ 1] [ 2]
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 .[ 2]
Jardine is known for his work on model category structures on simplicial presheaves .
^ "Full-time Faculty" . Department of Mathematics, University of Western Ontario . Retrieved February 11, 2018 .
^ a b "CV of Rick Jardine" . Department of Mathematics, University of Western Ontario . Retrieved February 11, 2018 .
External references [ edit ]