Graphs of Functions and Derivatives 129Polar Equations
The polar coordinate system locates points by a distance from the origin or pole, and an
angle of rotation. Points are represented by a coordinate pair (r,θ). If conversions between
polar and Cartesian representations are necessary, make the appropriate substitutions and
simplify.
x=rcosθ y=rsinθ r=√
x^2 +y^2 θ=tan−^1(y
x)Example 1
Convertr=4 sinθto Cartesian coordinates.
Step 1: Substitute inr=4 sinθto get
√
x^2 +y^2 =4 sin(
tan−^1
y
x)
.Step 2: Since sin
(
tan−^1
y
x)
=
y
√
x^2 +y^2, this becomes√
x^2 +y^2 = 4
y
√
x^2 +y^2.
Multiplying through by√
x^2 +y^2 , givesx^2 +y^2 = 4 y.Step 3: Complete the square onx^2 +y^2 − 4 y=0 to producex^2 +(y−2)^2 =4.
Example 2
Find the polar representation of
x^2
4
+
y^2
9=1.
Step 1: Substitute in
x^2
4
+
y^2
9
=1 to produce
(rcosθ)^2
4+
(rsinθ)^2
9=1.
Step 2: Simplify and clear denominators to get 9r^2 cos^2 θ+ 4 r^2 sin^2 θ=36, then factor for
r^2 (9 cos^2 θ+4 sin^2 θ)=36.
Step 3: Divide to isolater^2 =
36
9 cos^2 θ+4 sin^2 θ.
Step 4: Apply the Pythagorean identity to the denominatorr^2 =
36
5 cos^2 θ+ 4.
Types of Polar Graphs
SHAPE TYPICAL EQUATION NOTES
Line θ=k
Circle r=a Radius of the circle=a
r= 2 acosθ
r= 2 asinθ