We can’t return, we can only look behind from where we came, and go round and round and round in the circle game — “The Circle Game” by Joni Mitchell
Have you ever wondered why there are 360 degrees in a circle?
If today’s standards-making bodies had their way, there might be 100 degrees in a circle because we’re a base-10 society. But the number of degrees in a circle was devised before there was a standards body. It comes from the Babylonians (a base-60 society). They liked the number 60 because it is evenly divisible by a lot of numbers. That makes it easier to calculate fractions of 60. For example, half of 60 is 30, and a tenth of 60 is 6, etc. So, they were the first to designate 360 degrees — six sections of 60 degrees — as a complete circle.
Alternating current is closely related to the circle. The circular motion of a steam turbine drives an electric generator, which produces the voltage and current wave form. The key is the relative motion between a rotating magnetic field of the rotor and the coils of wire wound around the poles in the stator (also known as the armature). As the angle of the rotor changes, so does the amount of voltage generated in the armature. The resulting voltage varies smoothly and produces that familiar waveform we’ve all seen.
The relationship between the angle of the rotor and the value of the voltage is fixed. It all relates to the circle and to the right triangle. The ancient Babylonians recognized a fixed relationship between the three sides of a right triangle long ago. A tablet found dating back to 1,900 B.C. was translated as: 4 is the length and 5 is the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. And 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.
The tablet works out the 3–4–5 triangle, meaning that if a right triangle has one side that is 3 units long and another that is 4 units long, neither of which are the “diagonal,” then the diagonal has to be 5 units long to make all the lines connect. It uses a formula for calculating the third side of a right triangle, if you know the length of any two of the three sides. A Greek dude named Pythagoras later proved this formula, and it came to be known as the Pythagorean theorem. It simply states that there is a fixed relationship between the sides of a right triangle and it tells us what that relationship is.
There is a fixed relationship between one of the angles in a right triangle (one of the two not the 90-degree angle) and the ratio between the length of two of the sides. If we draw a series of right triangles, with one of the angles varying from zero to 90 degrees in increments of five degrees, and then measure two of the sides in each of the triangles, one side being opposite the angle and one being the diagonal, and expressed them as a ratio, we should get the following:
This table is the key to predicting the behavior of many cyclical phenomena. In the case of a generator and a sine wave, it gives us the answer to the fixed relationship between the angle of the rotor and value of the voltage. If we paint a picture of the angle between the orientation of the rotor relative to the orientation of the stator, then we can create a right triangle by drawing a vertical line to the base of the angle. Then, the length of the vertical line represents the magnitude (or the value) of the voltage. By looking at the chart, we can predict the voltage at any point around the circle. We need to know the peak voltage and multiply it by the ratio at the angle in question. This works because the vertical line represents the component of the movement that cuts the lines of flux in the magnetic field. The more obtuse the angle, the more lines of flux are cut by the armature and the more voltage that is generated.
If we continue the chart through 360 degrees and plot the voltage vs. the angle of the rotor, the result would look like this:
We could give this ratio a name and use it when we want to figure out anything relating to angles and circles. But fortunately, it already has a name, and it’s been around since 190 B.C., when a Greek named Hipparchus first compiled this table to help him track the motion of the sun and the moon. It’s called the sine function. The sine of the angle is the fixed ratio between the length of the line opposite the angle and the length of the diagonal line, or hypotenuse, as it is known.
Now that the cat is out of the bag, we can call a spade a spade and a sine a sine. You’ve just read an entire article about trigonometry, and hopefully, it wasn’t even painful. Do you love math yet? I do.
What’s your angle? E-mail the author at rcadena@plsn.com.
