**Problem : **
Do the rules of symmetry show that the following polar graph is symmeric
with respect to either the pole, the polar axis, or the line *θ* = ?

a) *r* = cos(*θ*) + 2.

b) *r* = 2 sin(*θ*).

c) *r* = 7.

d) *r* = 2 cos(3*θ*); e) *r* = .

a)

*r* = cos(*θ*) + 2 is symmetric with respect to the polar axis because

cos(*θ*) + 2 = cos(- *θ*) + 2.

b)

*r* = 2 sin(*θ*) is symmetric with
respect to the line

*θ* = because

2 sin(*θ*) = 2 sin(*Π* - *θ*).

c)

*r* = 7 is a circle -- it is symmetric with respect to the pole,
the polar axis, and the line

*θ* = .

d)

*r* = 2 cos(3*θ*) is symmetric with respect to the polar axis because

2 cos(3*θ*) = 2 cos(- 3*θ*).

e)

*r* = is symmetric with respect to the pole,
because

2 sin(2*θ*) = 2 sin(2(*θ* + *Π*)).

**Problem : **
What is the maximum value of | *r*| for the following polar equations: a) *r* = cos(2*θ*); b) *r* = 3 + sin(*θ*); c) *r* = 2 cos(*θ*) - 1.

a) The maximum value of

| *r*| in

*r* = cos(2*θ*) occurs when

*θ* = where

*n* is an integer and

| *r*| = 1.

b) The maximum value of

| *r*| in

*r* = 3 + sin(*θ*) occurs when

*θ* = +2*nΠ*
where

*n* is an integer and

| *r*| = 4.

c) The maximum value of

| *r*| in

*r* = 2 cos(*θ*) - 1 occurs when

*θ* = (2*n* + 1)*Π* where

*n* is an integer
and

| *r*| = 3.

**Problem : **
Find the intercepts and zeros of the following polar equations: a) *r* = cos(*θ*) + 1; b) *r* = 4 sin(*θ*).

a) Polar axis intercepts:

(*r*, *θ*) = (2, 2*nΠ*),(0,(2*n* + 1)*Π*), where

*n*
is an integer. Line

*θ* = intercepts:

(*r*, *θ*) = (1, + *nΠ*), where

*n* is an integer.

*r* = cos(*θ*) + 1 = 0 for

*θ* = (2*n* + 1)*Π*, where

*n* is an integer.

b) Polar axis intercepts:

(*r*, *θ*) = (0, *nΠ*) where

*n* is an integer. Line

*θ* =
intercepts:

(*r*, *θ*) = (4, +2*nΠ*) where

*n* is an integer.

*r* = 4 sin(*θ*) = 0 for

*θ* = *nΠ*, where

*n* is an integer.

**Problem : **
Decide whether each of the following polar graphs is a limacon, a rose
curve, a spiral, a circle, or none of these: a) *r* = 2 + cos(*θ*); b)
*r* = 2; c) *r* = sin(3*θ*); d) *r* = 1 - cos(*θ*); e) *r* = 2*θ*.

a)

*r* = 2 + cos(*θ*) is a limacon.

b)

*r* = 2 is a circle.

c)

*r* = sin(3*θ*) is a rose curve.

d)

*r* = 1 - cos(*θ*) is a limacon.

e)

*r* = 2*θ* is a spiral.