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Polar Equations, Polar Graphs, and Symmetry of Polar Graphs for AP Calculus

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By — McGraw-Hill Professional
Updated on Oct 24, 2011

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 x2 + y2 – 4y =0 to produce x2 +(y – 2)2 =4.

Example 2

Find the polar representation of

Step 1: Substitute in

Step 2: Simplify and clear denominators to get 9r2 cos2 θ +4r2 sin2 θ =36, then factor for r2(9 cos2 θ +4 sin2 θ)=36.

Step 3: Divide to isolate

Step 4: Apply the Pythagorean identity to the denominator

Types of Polar Graphs

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.

Types of Polar Graphs

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(–θ), symmetric about the line if f(θ)= f (πθ), and symmetric about the pole if 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(πθ) =2 – 4 cos θ, so the graph is not symmetric about the line θ = π2 .

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(πθ)=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(4θ).

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

Step 2:   5 cos(4(πθ))=5 cos(4π –4θ) which, by identity, is equal to 5[cos 4π cos 4θ + sin 4π sin θ] or 5 cos4θ, the graph is symmetric about the line θ =π/2.

Step 3:   Since 5 cos 4(θ + π)=5 cos(4θ + 4π)=5 cos(4θ), 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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