Precalculus by Richard Wright

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# 7-08 Graphs of Polar Equations

Summary: In this section, you will:

• Graph polar equations
• Identify symmetry in polar equations
• Find maximums and zeros of polar equations

SDA NAD Content Standards (2018): PC.5.2

All the planet and comets such as Phaethon, orbit the sun in elliptical orbits. A graph of their orbits can be made by plotting their positions over time and then connecting the points. Since ellipses are curves, it makes sense to use a coordinate system that is curved. This lesson is about graphing polar equations.

## Graph Polar Equations

Graph polar equations by making a table. The first column is r and the second column is θ. Choose values of θ and calculate the values of r. Then plot all the points. Make sure there are enough points to see the entire shape. Last, draw a smooth curve through the points.

###### Graph Polar Equations
1. Make table of $$\left(r, θ\right)$$.
2. Pick θ, calculate r.
3. Plot the points.
4. Draw a smooth curve through the points.

#### Example 1: Graph a Polar Equation

Graph $$r = 5 \sin θ$$.

###### Solution

Make a table for r and θ. Choose θ and calculate r.

 r θ 0 1.3 2.5 3.5 4.3 4.8 5 4.8 4.3 3.5 2.5 1.3 0 0 $$\frac{π}{12}$$ $$\frac{π}{6}$$ $$\frac{π}{4}$$ $$\frac{π}{3}$$ $$\frac{5π}{12}$$ $$\frac{π}{2}$$ $$\frac{7π}{12}$$ $$\frac{2π}{3}$$ $$\frac{3π}{4}$$ $$\frac{5π}{6}$$ $$\frac{11π}{12}$$ $$π$$
 r θ −1.3 −2.5 −3.5 −4.3 −4.8 −5 −4.8 −4.3 −3.5 −2.5 −1.3 0 $$\frac{13π}{12}$$ $$\frac{7π}{6}$$ $$\frac{5π}{4}$$ $$\frac{4π}{3}$$ $$\frac{17π}{12}$$ $$\frac{3π}{2}$$ $$\frac{19π}{12}$$ $$\frac{5π}{3}$$ $$\frac{7π}{4}$$ $$\frac{11π}{6}$$ $$\frac{23π}{12}$$ $$2π$$

Plot all the points and draw a nice curve. This graph is a circle.

Notice that the points in the second table repeated the points in the first table.

##### Try It 1

Graph $$r = 4 \cos \left(3θ\right)$$.

## Symmetry

Some graphs have symmetry and which makes allows you to use fewer points to make the graph. There are three types of symmetry: line $$θ = \frac{π}{2}$$ (y-axis), polar axis (x-axis), and pole (origin). Each of these has a test to determine if a polar equation has that type of symmetry.

###### Tests for Symmetry in Polar Equations

A graph has the following type of symmetry if a given substitution into the equation simplifies to the original equation.

• The line $$θ = \frac{π}{2}$$ (y-axis): Replace $$\left(r, θ\right)$$ with $$\left(r, π - θ\right)$$ or $$\left(-r, -θ\right)$$
• The polar axis (x-axis): Replace $$\left(r, θ\right)$$ with $$\left(r, -θ\right)$$ or $$\left(-r, π - θ\right)$$
• The pole (origin): Replace $$\left(r, θ\right)$$ with $$\left(r, π + θ\right)$$ or $$\left(-r, θ\right)$$

#### Example 2: Find Symmetry

Find the symmetry of $$r = 3\left(1 + \cos θ\right)$$.

###### Solution

Test for symmetry in the line $$θ = \frac{π}{2}$$. Substitute $$\left(r, π - θ\right)$$ for $$\left(r, θ\right)$$.

$$r = 3\left(1 + \cos θ\right)$$

$$r = 3\left(1 + \cos \left(π - θ\right)\right)$$

Use a difference formula ($$\cos\left(u - v\right) = \cos u \cos v + \sin u \sin v$$).

$$r = 3\left(1 + \left(\cos π \cos θ + \sin π \sin θ\right)\right)$$

$$r = 3\left(1 + \left(-\cos θ + 0\right)\right)$$

$$r = 3\left(1 - \cos θ\right)$$

This is not the same as the original, so it does not have $$θ = \frac{π}{2}$$ symmetry.

Test for symmetry in the polar axis. Substitute $$\left(r, -θ\right)$$ for $$\left(r, θ\right)$$.

$$r = 3\left(1 + \cos θ\right)$$

$$r = 3\left(1 + \cos \left(-θ\right)\right)$$

Use an even/odd identity ($$\cos\left(-u\right) = \cos u$$).

$$r = 3\left(1 + \cos θ\right)$$

This is the same as the original, so it does have polar axis symmetry.

Test for symmetry in the pole. Substitute $$\left(r, π + θ\right)$$ for $$\left(r, θ\right)$$.

$$r = 3\left(1 + \cos θ\right)$$

$$r = 3\left(1 + \cos \left(π + θ\right)\right)$$

Use a sum formula ($$\cos\left(u + v\right) = \cos u \cos v - \sin u \sin v$$).

$$r = 3\left(1 + \left(\cos π \cos θ - \sin π \sin θ\right)\right)$$

$$r = 3\left(1 + \left(-\cos θ + 0\right)\right)$$

$$r = 3\left(1 - \cos θ\right)$$

This is not the same as the original, so it does not have pole symmetry.

Just because the tests for symmetry do not simplify to the original equation, does not mean that symmetry does not exist. In general, polar equations that are functions of cosine are symmetric over the polar axis and functions of sine are symmetric over the line $$θ = \frac{π}{2}$$.

###### Quick Tests for Symmetry in Polar Equations

A graph has the following type of symmetry if a given substitution into the equation simplifies to the original equation.

• $$r = f\left(\sin θ\right)$$: Symmetric over line $$θ = \frac{π}{2}$$.
• $$r = f\left(\cos θ\right)$$: Symmetric over polar axis.
##### Try It 2

Find the symmetry of $$r = 3 + \sin \left(θ\right)$$.

###### Answer

This is a function of sine, so line $$θ = \frac{π}{2}$$. Also, it passes only the symmetry test for the line $$θ = \frac{π}{2}$$.

## Maximums and Zeros

Maximums of polar equations occur when |r| is at its largest. These are locations where the graph is farthest from the pole. Zeros occur when r = 0. These are locations where the graph is at the pole. Both maximums and zeros are found by looking for when the trigonometric portion of the equation is at a maximum or zero.

###### Maximums and Zeros of Polar Equations
• Maximums occurs when |r| is largest.
• Find angles where the trigonometric function is at its maximum.
• Zeros occur when r = 0.
• Find angles where the trigonometric function is 0.

#### Example 3: Find Maximums and Zeros

Find the maximums and zeros of $$r = 3 \sin 2θ$$.

###### Solution
• Maximums

The trigonometric portion is sin 2θ. Find the angles where it is at a maximum. The largest values of sine is 1 and −1.

sin 2θ = 1 when $$2θ = \frac{π}{2} + 2πn$$

sin 2θ = −1 when $$2θ = \frac{3π}{2} + 2πn$$

Divide both by 2.

$$θ = \frac{π}{4} + πn\text{ and }\frac{3π}{4} + πn$$

Plug in numbers for n until they reach 2π.

$$θ = \frac{π}{4}, \frac{3π}{4}, \frac{5π}{4}, \frac{7π}{4}$$

• Zeros

Let r = 0 and solve for θ.

$$0 = 3 \sin 2θ$$

$$0 = \sin 2θ$$

$$2θ = 0 + πn$$

$$θ = \frac{π}{2}n$$

$$θ = 0, \frac{π}{2}, π, \frac{3π}{2}$$

##### Try It 3

Find the maximums and zeros of $$r = 5 \cos 3θ$$.

###### Answer

Maximums: $$θ = 0, \frac{π}{3}, \frac{2π}{3}, π, \frac{4π}{3}, \frac{5π}{3}$$; Zeros: $$θ = \frac{π}{6}, \frac{π}{2}, \frac{5π}{6}, \frac{7π}{6}, \frac{3π}{2}, \frac{11π}{6}$$

## Special Graphs of Polar Equations

There are several special polar graphs. These are fairly simple in polar form, but very complicated in rectangular form.

• Limaçons: $$r = a ± b \cos θ$$ or $$r = a ± b \sin θ$$ where a > 0 and b > 0.

• Rose Curves: n petals if n is odd; 2n petals if n is even; n ≥ 2

• Circles and Lemniscates:

##### Lesson Summary

###### Graph Polar Equations
1. Make table of $$\left(r, θ\right)$$.
2. Pick θ, calculate r.
3. Plot the points.
4. Draw a smooth curve through the points.

###### Tests for Symmetry in Polar Equations

A graph has the following type of symmetry if a given substitution into the equation simplifies to the original equation.

• The line $$θ = \frac{π}{2}$$ (y-axis): Replace $$\left(r, θ\right)$$ with $$\left(r, π - θ\right)$$ or $$\left(-r, -θ\right)$$
• The polar axis (x-axis): Replace $$\left(r, θ\right)$$ with $$\left(r, -θ\right)$$ or $$\left(-r, π - θ\right)$$
• The pole (origin): Replace $$\left(r, θ\right)$$ with $$\left(r, π + θ\right)$$ or $$\left(-r, θ\right)$$

###### Quick Tests for Symmetry in Polar Equations

A graph has the following type of symmetry if a given substitution into the equation simplifies to the original equation.

• $$r = f\left(\sin θ\right)$$: Symmetric over line $$θ = \frac{π}{2}$$.
• $$r = f\left(\cos θ\right)$$: Symmetric over polar axis.

###### Maximums and Zeros of Polar Equations
• Maximums occurs when |r| is largest.
• Find angles where the trigonometric function is at its maximum.
• Zeros occur when r = 0.
• Find angles where the trigonometric function is 0.

Helpful videos about this lesson.

## Practice Exercises

Identify the type of graph.

1. Identify the symmetry of the polar graphs.

2. $$r = 3 - 4 \cos θ$$
3. $$r = 4 \cos θ$$
4. $$r = 1 + \sin θ$$
5. Find the maximum values of |r| and the zeros of r.

6. $$r = 2 + \cos θ$$
7. $$r = 4 \sin θ$$
8. $$r = 3 \cos 2θ$$
9. Graph the polar equations.

10. $$r = 4 \cos θ$$
11. $$r = 4 \sin θ$$
12. $$r = 2 + \cos θ$$
13. $$r = 3 \cos 2θ$$
14. $$r = 2 - 3 \sin θ$$
15. Use a graphing utility to graph the polar equations.

16. $$r = 5 \sin 4θ$$
17. $$r = 1 + \cos θ$$
18. Mixed Review

19. (7-07) Convert the point to rectangular coordinates: $$\left(2, \frac{5π}{4}\right)$$.
20. (7-07) Convert the equation to rectangular coordinates: $$r = 2 \sin θ$$.
21. (7-06) Graph \left\{\begin{align}x &= 2t\\y &= \frac{1}{2}t^3\end{align}\right..
22. (7-05) Classify the conic: $$3x^2 - xy + 4y^2 - 12x + 3y + 2 = 0$$.
23. (7-03) Find the standard equation of the ellipse with vertices (±5, 0) and foci (±3, 0).

### Answers

1. Rose curve with 7 petals
2. Cardioid
3. polar axis
4. polar axis
5. line $$θ = \frac{π}{2}$$
6. Maximums: θ = 0, π; Zeros: none
7. Maximums: $$θ = \frac{π}{2}, \frac{3π}{2}$$; Zeros: θ = 0, π
8. Maximums: $$θ = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$$; Zeros: $$\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$
9. $$\left(-\sqrt{2}, -\sqrt{2}\right)$$
10. $$x^2 + \left(y - 1\right)^2 = 1$$
11. Ellipse
12. $$\frac{x^2}{25}+\frac{y^2}{16} = 1$$