We need to sketch *r* = 3 cos 2*θ*.

Let's find the answer using the following interactive graph. You will trace out the required curve as you change the angle.

**Drag** the blue dot (slowly) left and right to change the angle *θ* and observe the resulting changes in the value of *r*.

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**Hint:** Let go of the blue dot to see a smooth curve at any time. You can trace out positive or negative angles.

[In the above graph, angles are in **radians**, where *π* radians = 180°. To learn more, see: Radians.]

Notice the curve is fully drawn once θ takes all values between 0 and 2*π*.

What if you can't use a computer to draw the graph?

You'll need to set up a table of values, as follows. I've put degrees and the radian equivalents.

θ (degrees) | 0° | 30° | 60° | 90° | 120° | 150° | 180° |
---|---|---|---|---|---|---|---|

θ (radians) | `0` | `π/6` | `π/3` | `π/2` | `(2π)/3` | `(5π)/6` | `π` |

`r = 3\ cos\ 2θ` |
`3` | `1.5` | `-1.5` | `-3` | `-1.5` | `1.5` | `3` |

θ (degrees) | 210° | 240° | 270° | 300° | 330° | 360° |
---|---|---|---|---|---|---|

θ (radians) | `(7π)/6` | `(4π)/3` | `(3π)/2` | `(5π)/3` | `(11π)/6` | `2π` |

`r = 3\ cos\ 2θ` |
`1.5` | `-1.5` | `-3` | `-1.5` | `1.5` | `3` |

The first 7 points from this table are (3, 0°), (1.5, 30°), (-1.5, 60°), (-3, 90°), (-1.5, 120°), (1.5, 150°), and (3, 180°).

Placing those first 7 points on a polar coordinate grid gives us the following:

We start at Point 1, (3, 0°), and move around the graph by increasing the angle and changing the distance from the origin (determined by substituting the angle into *r* = 3 cos 2*θ*. I have drawn arrows to indicate the basic direction we have to head in to get to the next point.

**Recall: **A negative "*r*" means we need to be on the opposite side of the origin.

I have only plotted the first 7 points above to keep the graph simple. Clearly, we would need to calculate more than this number of points to get a good sketch. (You would need at least twice as many points as I have in the table above - every 15° would be adequate.)

Here's the complete graph.

Graph of *r* = 3 cos (2θ).

Next, here's the answer for the conversion to rectangular coordinates.

**Why? **We convert the function given in this question to rectangular coordinates to see how much simpler it is when written in polar coordinates.

To convert `r = 3\ cos\ 2θ` into **rectangular coordinates**, we use the fact that

cos 2

θ= cos^{2}θ- sin^{2}θ.

So *r* = 3 cos 2*θ* = 3(cos^{2 }*θ* - sin^{2} *θ*).

Now since `cos\ theta=x/r`, `sin\ theta=y/r` and `r^2=x^2+y^2`, we have

`cos^2theta=(x^2)/(r^2)=(x^2)/(x^2+y^2)`

and

`sin^2theta=(y^2)/(r^2)=(y^2)/(x^2+y^2)`

Taking the positive square root of `r^2=x^2+y^2` gives us:

`r=sqrt(x^2+y^2)`

So `r = 3 cos\ 2θ` in polar coordinates is equivalent to

`r=3(cos^2theta-sin^2theta)`

`sqrt(x^2+y^2)=3(x^2/(x^2+y^2)-y^2/(x^2+y^2))`

`sqrt(x^2+y^2)=3/(x^2+y^2)(x^2-y^2)`

`(x^2+y^2)^(3"/"2)=3(x^2-y^2)`

in rectangular coordinates.

We see that our equation in polar coordinates, *r* = 3 cos 2*θ*, is much simpler than the rectangular equivalent.