Polar Coordinate

 

From elementary to high school, you have learned to plot a simple point on what is known as a Cartesian plane. For example, when plotting (2, 3), we would move 2 units to the right off the x-axis and 3 units up:







But now, the squares and straight lines you focused on in algebra are transitioning to curves and circles in trigonometry!
PRESENTING: THE POLAR GRAPH!

Polar graphs have circles and angles, so instead of the normal x or y-axis, these use DEGREES and
RADII. The coordination is as follows:

                   (radius, Θ) or (Θ, radius)

The rule for plotting points is very simple:
you can either
-turn to the direction and move radius units towards it or

-move radius units and walk to the direction


EXAMPLES:

EXAMPLE: Plotting Points

Plot the following points: (5, 330º), (2, 180º), and (-45º, 2)

·         (2,180°): Move forward 5, turn 330 degrees

·         (-45°, 2): Move forward 2, turn negative 45 degrees

·         (2,180°): Turn 180 degrees, move forward 2

**Note, there is no one exactly way to plot these points, you can choose any method you like to get to the destination.                                                                                                                                                                     Imagine a Cartesian graph on top of a Polar graph. Where would the point (4, 60°) be on the Cartesian graph?

HINT: Try imagining the 0° and 90° as the x and y-axis.
To find the points on a Cartesian plane, we must find how many units, a,
it moves on the x-axis, and how many b-units are moved on the y-axis.
 



EXAMPLE: Polar >> Rectangular
                                                                                                                                                    Where's the polar point (4, 60°) on a Cartesian graph?

           <<<< Here's what we have so far

Because this is a right triangle, we can now use our trig to solve for the
triangle's sides!doesn't this look familiar *cough* vectors *cough*
1) sin(60°) = b/4                     1) cos(60°) = a/4
2) 4sin(60°) = b                      2) 4cos(60°) = a
3) 4(√3/2) = b                         3) 4(1/2) = a
4) 2√3 = b                               4) 2 = a
The polar point (4, 60°) would be exactly where (2, 2√3) would be on a Cartesian graph.

Now, where would (3, 4) be on a polar graph?

EXAMPLE: Rectangular >> Polar

Because this is a right triangle, we can now use our trig to solve for the angle (Ɵ) and radius (r)!

1) tan Ɵ = 4/3                                                                                        1) r² = 3² + 4²
2) Ɵ = tan‾ ¹(4/3)                                                                                   2) r² = 9 + 16
3) Ɵ = 53.13°                                                                                         3) r² = 25
                                                                                                              4) r = 5



The coordinate point (3,4) would be exactly
(5, 53.13°) on a polar graph. That's it!

-TrigInADay