footnotes:
(1):
table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/12]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)
+----------+------------+--------------+
| $i$ | $cos(2*i)$ | $rational$ |
+==========+============+==============+
| 0 | 1.000 | 1 |
+----------+------------+--------------+
| 1/12*pi | 0.8660 | 1/2*sqrt(3) |
+----------+------------+--------------+
| 1/6*pi | 0.5000 | 1/2 |
+----------+------------+--------------+
| 1/4*pi | 0.0000 | 0 |
+----------+------------+--------------+
| 1/3*pi | -0.5000 | -1/2 |
+----------+------------+--------------+
| 5/12*pi | -0.8660 | -1/2*sqrt(3) |
+----------+------------+--------------+
| 1/2*pi | -1.000 | -1 |
+----------+------------+--------------+
| 7/12*pi | -0.8660 | -1/2*sqrt(3) |
+----------+------------+--------------+
| 2/3*pi | -0.5000 | -1/2 |
+----------+------------+--------------+
| 3/4*pi | 0.0000 | 0 |
+----------+------------+--------------+
| 5/6*pi | 0.5000 | 1/2 |
+----------+------------+--------------+
| 11/12*pi | 0.8660 | 1/2*sqrt(3) |
+----------+------------+--------------+
| pi | 1.000 | 1 |
+----------+------------+--------------+
| 13/12*pi | 0.8660 | 1/2*sqrt(3) |
+----------+------------+--------------+
| 7/6*pi | 0.5000 | 1/2 |
+----------+------------+--------------+
| 5/4*pi | 0.0000 | 0 |
+----------+------------+--------------+
| 4/3*pi | -0.5000 | -1/2 |
+----------+------------+--------------+
| 17/12*pi | -0.8660 | -1/2*sqrt(3) |
+----------+------------+--------------+
| 3/2*pi | -1.000 | -1 |
+----------+------------+--------------+
| 19/12*pi | -0.8660 | -1/2*sqrt(3) |
+----------+------------+--------------+
| 5/3*pi | -0.5000 | -1/2 |
+----------+------------+--------------+
| 7/4*pi | 0.0000 | 0 |
+----------+------------+--------------+
| 11/6*pi | 0.5000 | 1/2 |
+----------+------------+--------------+
| 23/12*pi | 0.8660 | 1/2*sqrt(3) |
+----------+------------+--------------+
| 2*pi | 1.000 | 1 |
+----------+------------+--------------+
To plot: t = var('t')
polar_plot(cos(2*t), t, 0, 2*pi)
sage: table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)
+---------+------------+--------------+
| $i$ | $cos(2*i)$ | $rational$ |
+=========+============+==============+
| 0 | 1.000 | 1 |
+---------+------------+--------------+
| 1/8*pi | 0.7071 | 1/2*sqrt(2) |
+---------+------------+--------------+
| 1/4*pi | 0.0000 | 0 |
+---------+------------+--------------+
| 3/8*pi | -0.7071 | -1/2*sqrt(2) |
+---------+------------+--------------+
| 1/2*pi | -1.000 | -1 |
+---------+------------+--------------+
| 5/8*pi | -0.7071 | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/4*pi | 0.0000 | 0 |
+---------+------------+--------------+
| 7/8*pi | 0.7071 | 1/2*sqrt(2) |
+---------+------------+--------------+
| pi | 1.000 | 1 |
+---------+------------+--------------+
| 9/8*pi | 0.7071 | 1/2*sqrt(2) |
+---------+------------+--------------+
| 5/4*pi | 0.0000 | 0 |
+---------+------------+--------------+
| 11/8*pi | -0.7071 | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/2*pi | -1.000 | -1 |
+---------+------------+--------------+
| 13/8*pi | -0.7071 | -1/2*sqrt(2) |
+---------+------------+--------------+
| 7/4*pi | 0.0000 | 0 |
+---------+------------+--------------+
| 15/8*pi | 0.7071 | 1/2*sqrt(2) |
+---------+------------+--------------+
| 2*pi | 1.000 | 1 |
+---------+------------+--------------+
sage: table([(i,n(sin(2*i), digits=4), sin(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$sin(2*i)$", "$rational$"], frame=True)
+---------+------------+--------------+
| $i$ | $sin(2*i)$ | $rational$ |
+=========+============+==============+
| 0 | 0.0000 | 0 |
+---------+------------+--------------+
| 1/8*pi | 0.7071 | 1/2*sqrt(2) |
+---------+------------+--------------+
| 1/4*pi | 1.000 | 1 |
+---------+------------+--------------+
| 3/8*pi | 0.7071 | 1/2*sqrt(2) |
+---------+------------+--------------+
| 1/2*pi | 0.0000 | 0 |
+---------+------------+--------------+
| 5/8*pi | -0.7071 | -1/2*sqrt(2) |
+---------+------------+--------------+
| 3/4*pi | -1.000 | -1 |
+---------+------------+--------------+
| 7/8*pi | -0.7071 | -1/2*sqrt(2) |
+---------+------------+--------------+
| pi | 0.0000 | 0 |
+---------+------------+--------------+
| 9/8*pi | 0.7071 | 1/2*sqrt(2) |
+---------+------------+--------------+
| 5/4*pi | 1.000 | 1 |
+---------+------------+--------------+
| 11/8*pi | 0.7071 | 1/2*sqrt(2) |
+---------+------------+--------------+
| 3/2*pi | 0.0000 | 0 |
+---------+------------+--------------+
| 13/8*pi | -0.7071 | -1/2*sqrt(2) |
+---------+------------+--------------+
| 7/4*pi | -1.000 | -1 |
+---------+------------+--------------+
| 15/8*pi | -0.7071 | -1/2*sqrt(2) |
+---------+------------+--------------+
| 2*pi | 0.0000 | 0 |
+---------+------------+--------------+
___________________________________________________________
(2):
How tables can be generated in Sage to help plot polar graphs. Now you can copy and paste it into
http://sagecell.sagemath.org to see what I mean:
table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/12]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)
___________________________________________________
t = var('t')
p1 = polar_plot(cos(2*t), t, 0, 2*pi)
p2 = polar_plot(sin(2*t), t, 0, 2*pi, linestyle="--")
show(p1 + p2)
_____________________________________________
table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)