Calculus: Analytic Geometry and Calculus, with Vectors

530 Polar, cylindrical, and spherical coordinates

two of the formulas can be replaced by the single formula 4) = tan-' ylx.

These formulas enable us to make substitutions which transform formulas

involving coordinates of one brand into formulas involving coordinates

of the other brand.

The remainder of the text of this section is devoted to development of

the art of sketching polar coordinate graphs of given equations for which

it is more or less appropriate to operate without restricting p to non-

negative values and without restricting 95 to an interval such as the

interval -r < 4 < r. In this situation, the relation between polar

graphs and equations is complicated by the fact that each point has an

infinite set of polar coordinates and we need a definition. The polar

coordinate graph of an equation of the form f(p,o) = 0 is the set S con-

sisting of points P each of which has at least one set (p,4)) of polar coordi-

nates for which f(p,o) = 0. The polar graph of the equation p = -1

is then the unit circle C with center at the origin because PP(-1,4)) is,

for each 4), the same as the point P,,(1, 0 + r). The polar graphs of the

Figure 10.143

two equations p = -1 and p = 1 therefore coincide,

even though there is clearly no single pair (p,4)) of

numbers for which the two equations are simultane-

ously satisfied. Figure 10.143 can promote under-

standing of this matter. When we allow p to be

negative, the polar graph of the equation 0 = 0 is

more than the initial half-line of the polar coordinate

system; it is the entire line upon which the initial

half-line lies.

Supposing that a is a given positive constant, we undertake to deter-

mine the nature of the polar graph of the equation

(10.15) p = a sin 20

without laboriously locating many points. Observing that IpI < a

always and Jpl = a sometimes, we draw the circle of radius a with center

at the origin to help us. Our knowledge of the sine tells us that sin 20

increases from 0 to 1 and p increases from 0 to a as 20 increases from 0

to 7/2 and hence as 0 increases from 0 to v/4. This information enables

us to sketch the first part of the first leaf, or loop, of Figure 10.151.

Similarly, p decreases from a to 0 as 24) increases from r/2 to r and hence

as 0 increases from r/4 to r/2. Now we complete the first leaf. Con-

tinuing to decrease, p decreases from 0 to -a as 20 increases from r to

3r/2 and hence as 0 increases from r/2 to 37/4. During this operation,

The terminal side of the angle ¢ is in the second quadrant and negativeness

of p throws the graph into the fourth quadrant to give the first half of

the second leaf. Then p increases from -a to 0 as 20 increases from

3r/2 to 2r and hence as 0 increases from 3r/4 tor. This gives the

second half of the second leaf. Continuing its increase, p increases from