Trigonometric method:
The pictorial method is inaccurate. It can be a bother, at times, to divide down and get numbers which are easy to work with. Drawing pictures accurately requires care. If the phase angle is close to 0° or 90°-that is, if the ratio XL /R is quite small or large-the accuracy is poor in drawing.
There is a better way. If you have a calculator which can find the arctangent of a number, you have got it made easy. Nowadays, if you intend to work with the engineers, there is no excuse not to possess a good calculator. If you do not have one, I go out and buy one right now. It should have trig functions and the inverses of them, log functions, exponential functions, and others which engineers use often.
If you know values XL and R, then phase angle is simply the arctangent of their ratio, or arctan (XL /R). This can be written tan-1 (XL /R). Punch a few buttons on the calculator.
Figure-- Another pictorial way of finding phase angle. This method shows actual impedance vector.
Problem:
The inductive reactance in the RL circuit is given 680 Ω, and the resistance is 840 Ω. What is phase angle of it?
Find ratio XL/R = 680/840. The calculator will display something around 0.809523809. Find the arctangent, or tan-1, getting the phase angle of 38.99099404 degrees. Round this off to 39.0 degrees.