Plotting polar coordinates requires two steps. First, determine the angle, then determine the distance away from the origin.
Coterminal angles play a big role in Polar Coordinates because θ can be positive or negative.
Recall that positive angles a plotted counter-clockwise and negative angles are clockwise.
Note that θ and -θ have the same acute reference angle and that they always terminate in the same half of the x-axis. Therefore, they have the same Cosine values. However, their Sine values will be opposites.
This is reflected in the following theorem:
Recall from our study of functions that Cosine is an Even function and Sine is Odd.
Step 1: Plot the angle
Plot the distance away from the origin, the 'r' value
Fortunately, all this means is that the 'r' component is plotted in the opposite direction across the origin.
Converting from Polar Coordinates to Rectangular
When we want to convert from Polar to rectangular coordinates, all we need are x and y values. These correspond to rCos(θ) and rSin(θ)
Converting from Rectangular to Polar Coordinates
This is a little bit more complicated because we need two values: the 'r' (radius) and angle Theta
Finding 'r' is easy. It's just the Pythagorean Theorem.
Finding θ is also pretty easy. It involves the Inverse Tangent, but you do have to check the quadrant to make sure the angle is correct.