## Wednesday, March 30, 2011

### COMPLEX NUMBERS...TO POLAR COORDINATES...TO RETANGULAR COORDINATES

What’s a COMPLEX NUMBER? We know about Real Numbers, Imaginary Numbers, and Undefined Numbers, but these are all relatively simple and stand alone. What if we combine a Real Number and an Imaginary one through addition or subtraction? (Sometimes using the Quadratic Formula gives this type of answer.) It’s a little more complicated and so “complex” is a good descriptor.

To graph the complex number on a coordinate plane, the X-axis represents the Real Number and the Y-axis represents the coefficient of the Imaginary Number.

There’s another way to measure where point (3 + 4i) is on the coordinate graph. What if we measured the angle around the unit circle and then how long the ray is from the origin? This would give us POLAR COORDINATES.

Oh, look!! A right triangle!! ( Thank you, Ms. Smith, your 9th grade geometry teacher who made sure everyone knew the Pythagorean Theorem very well! Not to mention SohCahToa!)

To find the angle between the ray and the x-axis, we could use Tangent.

The opposite side of the triangle is the coefficient on the imaginary number (b) and the adjacent side is the Real Number component (a) of the Complex Number.

To find the length of the ray, we use the Pythagorean Theorem:

where a and b come from the complex number and c is the length of the ray.

From our complex number example, a = 3, b = 4, and c = 5, while θ = 53.13 degrees or .93 radians.

So, we could locate the starred point on the graph by using POLAR COORDINATES.

In polar coordinates, we aren’t looking for the intersection of X=some number and Y= some number, but where is the angle located on the Unit Circle and how long is the ray extending from the origin. The ordered pair is

Read this ordered pair..."are, theda".......where have you heard this “phrase” before? Remember angular velocity? The length of the arc on a circle is s=rθ. [If you are graphing Polar Coordinates, recognize a NEGATIVE 'r' starts at the origin but moves in the OPPOSITE DIRECTION and a NEGATIVE theda moves in a clockwise direction. Can you think of 4 different ordered pairs to indicate (4,π) on the Polar Graph using negatives?]

We could also describe the location of the star by using TRIGONOMETRIC COORDINATES.

For this transformation, we should remember from Trigonometry (or preview if you haven’t taken Trig yet) that the y-axis represents Sine and the x-axis represents Cosine.

These are, in fact, the equations you would use to convert from Polar Coordinates to Rectangular Coordinates.

Here’s a typical problem:

Convert the complex number to Polar Coordinates and Rectangular Coordinates.

[Print out this reference box and add it to your review card.]

THINKING QUESTIONS:
1. How would subtracting the imaginary number effect the placement of a point on the graph?

2. How would this change the Polar and Rectangular Coordinates?

3. How do you use inverse trig functions and reference angles to find angles not in quadrant I?

Doesn’t this stuff make you think in CIRCLES? Or maybe it just makes your mind go round and round. GOOD! Because next we’ll be looking at Cardioids and Limacons (with and without interior loops). If you want to see a really impressive, interactive display of the new concept, go to

http://www.intmath.com/plane-analytic-geometry/ans-8.php?a=1

and be amazed!!