534 Polar, cylindrical, and spherical coordinates
9 Sketch polar graphs of the equations(a)p=3+2cos¢ (b)p=3+4cos4>
(c) p = a sec ¢ (d) p = tan 95
(e) p = a sin 30 (f) p = a sin 3(g) P= 1/4>,(4> > 0) (h)P = 1/(1 + 4,2),(4> > 0)
10 Sketch rectangular graphs of y = cos x, y = cos x, and y = cost x in
one figure, and then sketch polar graphs of p = cos (k, p = cos ¢, and p =
cost 0 in another figure. Partial solution: Good polar graphs of p = cos di and
p = cos 0 and the right-hand half of the graph of p = cost ¢ areshown in
Figure 10.191.
90° 80° 70°_90° _80° _70°
Figure 10.191
11 Sketch rectangular graphs of x2 + y2 = 1 and x2 + y2 = 25 in one figure
and then sketch polar graphs of p2 + 02 = 1 and p2 + 02 = 25 in another figure.
12 In case pI > 0 and p2 > 0, a straightforward application of the law of
cosines gives the formulad2 = Pi + ps - 2PIP2 cos (02 - 01)
for the square of the distance d between points having polar coordinates (pl,4>i)
and (p2,42). Show that the formula is also valid when p, S 0 or P2 5 0 or both.
13 This is a rather heroic problem that requires careful and accurate applica-
tions of rules for differentiation and attention to algebraic details. No vectors
or figures or tricks of any kind are to be used; just differentiate and substitute.
Using the formulas
x(t) = p(t) cos 4>(t), y(t) = p(t) sin ¢(t)