Calculus-Sec-10-3-Solution
10.3 Polar Coordinates by SGLee - HSKim - JHLee
1-4. Plot the point whose polar coordinates are given. Then, find the Cartesian coordinates of the point.
1. 
= 
http://matrix.skku.ac.kr/cal-lab/cal-9-3-1.html
http://matrix.skku.ac.kr/cal-lab/sage-grapher-polar.html
2. 
= 
3. 
= 
.
4. 
=
.
5-8. The Cartesian coordinates are given.
Find two other pairs of polar coordinates of the point, one with 
and other 
.
5. 
, and 
.
Since 
is in the fourth quadrant, the polar coordinates are
⑴ 
: 
, ⑵ 
: 
.
6. 
, and 
.
Since 
is in the second quadrant, the polar coordinates are
⑴ 
: 
, ⑵ 
: 
.
7. 
, and 
.
Since 
is in the fourth quadrant, the polar coordinates are
⑴ 
: 
, ⑵ 
: 
.
8. 
, and 
.
Since 
is in the second quadrant, the polar coordinates are
⑴ 
: 
, ⑵ 
: 
.
9. Find a formula for the distance between the points with polar coordinates 
and 
.
Let 
Let 
be the e distance between 
and 
.
10-11. Find a polar equation for the curve represented by the given Cartesian equation.
10. 
.
11. 
.
12. 
.
13-14. Find a Cartesian equation for the curve represented by the given polar equation.
13. 
,
a circle of radius 
centered at 
.
The first two equations are actually equivalent since
.
But 
gives the point 
when 
.
Thus, the single equation 
is equivalent to the compound condition (
).
14. 
.
15. 
16-25. Sketch the curve with the given polar equation.
16. 
24. 
25. 
26. Show that the polar curve 
(called a conchoid) has the line 
as a vertical asymptote by showing that 
. Use this fact to sketch the conchoid.
http://matrix.skku.ac.kr/cal-lab/cal-9-3-26.html
.
Now, 
,
so 
.
Also, 
, so
.
Therefore, 
is a vertical asymptote.
27. Show that the curve 
(also a conchoid) has the line 
as a horizontal asymptote by showing that 
. Use this fact to help sketch conchoid.
http://matrix.skku.ac.kr/cal-lab/cal-9-3-27.html
.
Now, 
,
so 
.
Also, 
,
so 
.
Therefore, 
is a horizontal asymptote.
28. Show that the curve 
(called a cissoid of Diocles) has the line 
as a vertical asymptote. Show also that the curve lies entirely within the vertical strip 
.
Use these facts to sketch the cissoid.
29-32. Find the slope of the tangent line at the given point.
29. 
.
30. 
.
31. 
.
32. 
33-36. Find the points on the given curve where the tangent line is horizontal or vertical.
33. 
.
.
So the tangent is vertical at 
and 
.
. So the tangent is horizontal at 
and 
.
34. 
35. 
36. 
37. Show that the polar equation 
represents a circle and find its center and radius.
.
Then the polar equation 
represents a circle and its center : 
, radius :
.
17. 
.
18. 
19. 
21. 
22. 
, 
, 
