Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

198

ANALYTIC GEOMETRY

6. r cos 0=4.
8. rcos 0 =- 4.
10. rsin0 4 r cos 0 = 3.
12. 4rsin0+3rcos90=5.
14. tan0=-1.
16. 0 = 60~.

7. rsin =4.
9. rsin=- 4.
11. r sin 0 - r cos 0 = 3.
13. 5r sin0 - 12rcos 0= 26.
15. 2 cos 0 =0.
17. 0 180~.

4. Graphs of Equations. If an equation in polar coordinates
is given, which cannot be reduced to one of the forms recognized as representing a known curve, it is necessary, in order
to determine what curve is defined by the equation, to plot a
reasonable number of points whose coordinates satisfy the
equation. But considerations of symmetry will often shorten
the work.
Example 1. Consider the equation

(1) r2 = 1
This equation is equivalent to
(2) r=4

L6 sin 0.

Lt/sin 0,

where we have taken only the positive square root, since negative values of r have for us no meaning.
When 0 = 0, r = 0; as 0 increases, r increases, and when
0 = 90~, r= 4. Using a table of sines and a table of square
roots, we compute the following coordinates of further points
of the curve.

0 10~ 20~
r 1.67 2.34

30~ 400 50~ 60~ 700 80~
2.83 3.21 3.50 3.72 3.88 3.97
More computations are unnecessary.
For, the curve is symmetric in the ray
0 = 90~. To prove this, we note that, if
P: (r, 0) is any point of the curve, then
the point P': (r, 180~ —0), which is
symmetric to P in the ray 0= 90~ is
also a point of the curve, inasmuch as

0
FIG. 8

sin (180~ - 0) - sin 0.