Practice problems for these concepts can be found at: Graphs of Functions and Derivatives Practice Problems for AP Calculus

### Polar 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.

#### Example 1

Convert *r* =4 sin θ to Cartesian coordinates.

Step 1: Substitute in *r* =4 sin θ to get .

Step 2: Since sin Multiplying through by .

Step 3: Complete the square on *x*^{2} + *y*^{2} – 4y =0 to produce *x*^{2} +(*y* – 2)^{2} =4.

#### Example 2

Find the polar representation of

Step 1: Substitute in

Step 2: Simplify and clear denominators to get 9*r*^{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 isolate

Step 4: Apply the Pythagorean identity to the denominator

### Types of Polar Graphs

#### Example 1

Classify each of the following equations according to the shape of its graph.

(a) *r* =5+7 cosθ, (b) , (c) *r* = 4 – 4 sin θ.

The equation in (a) is a limaç, and since < 1, 1, it will have an inner loop. The equation in (b) is a spiral. Equation (c) appears at first glance to be a limaç; however, since the coefficients are equal, it is a cardiod.

#### Example 2

Sketch the graph of *r* =3 cos(2θ). The equation *r* = 3 cos (2θ) is a polar rose with four petals each 3 units long. Since 3 cos(0) = 3, the tip of a petal sits at 3 on the polar axis.

### Symmetry of Polar Graphs

A polar curve of the form *r* = *f( θ)* will be symmetric about the polar, or horizontal, axis if

*f(*=

*θ*)*f(–*, symmetric about the line if

*θ*)*f*(

*)=*

*θ**f*(

*–*

*π**), and symmetric about the pole if*

*θ**f*(

*)=*

*θ**f*(

*+*

*θ**).*

*π*#### Example 1

Determine the symmetry, if any, of the graph of *r* =2 + 4 cos *θ*.

Step 1: Since 2 + 4 cos(–*θ*) =2 + 4 cos *θ*, the graph is symmetric about the polar axis.

Step 2: 2 + 4 cos(*π* – *θ*) =2 – 4 cos *θ*, so the graph is not symmetric about the line *θ* = *π*2 .

Step 3: Since 2 + 4 cos(*θ* + *π*) =2 + 4 [cos *θ* cos *π* – sin *θ* sin *π*] =2–4 cos *θ*, the graph is not symmetric about the pole.

#### Example 2

Determine the symmetry, if any, of the graph of *r* = 3 – 3 sin *θ*.

Step 1: Since 3 – 3 sin(–*θ*) =3 + 3 sin *θ* is not equal to r =3 – 3 sin *θ*, the graph is not symmetric about the polar axis.

Step 2: 3 – 3 sin(*π* – *θ*)=3 – 3 sin *θ*, so the graph is symmetric about the line *θ* =*π*/2.

Step 3: Since 3 – 3 sin(*θ* + *π*)=3 – 3[sin *θ* cos *π* + sin *π* cos *θ*]=3 + 3 sin *θ*, the graph is not symmetric about the pole.

#### Example 3

Determine the symmetry, if any, of the graph of *r* =5 cos(4*θ*).

Step 1: Since 5 cos(4(–*θ*))=5 cos 4*θ*, the graph is symmetric about the polar axis.

Step 2: 5 cos(4(*π* –*θ*))=5 cos(4*π* –4*θ*) which, by identity, is equal to 5[cos 4*π* cos 4*θ* + sin 4*π* sin *θ*] or 5 cos4*θ*, the graph is symmetric about the line *θ* =*π*/2.

Step 3: Since 5 cos 4(*θ* + *π*)=5 cos(4*θ* + 4*π*)=5 cos(4*θ*), the graph is symmetric about the pole.

Practice problems for these concepts can be found at: Graphs of Functions and Derivatives Practice Problems for AP Calculus

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