Polar Coordinates to Rectangular Coordinates Conversion
Converting from rectangular to polar coordinates is another practical application of trigonometry.
If you assume that the "R" coordinate is the horizontal axis of a right triangle and that the "X" coordinate is the vertical axis of a right triangle then the resultant hypotenuse of this triangle is the Impedance.
( Z = R + X )
The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two remaining sides of that right triangle.
(C x C) = (A x A) + (B x B)
or in the specific case of finding Impedance (Z):
(Impedance x Impedance ) = (Resistance x Resistance) + (Reactance x Reactance)
(Z x Z) = (R x R) + (X x X)
The Impedance Angle can easily be found by using Trigonometric functions to find the "Angle Theta".
(Angle Theta is the angle above the horizontal.)
Sine, Cosine or Tangent functions are easily applied depending on which sides you desire to work with.
The Arc Tangent of the ratio of Reactance divided by the Resistance will yield the Impedance angle.
Angle of Z = arc tan(X/R).
Converting from polar coordinates to rectangular is more simple trig.
R = Z x cos.(angle)
X = Z x sin(angle)