﻿VK5AJL - Geometry of waves - Sin(e) wave

WAVE GEOMETRY - WHAT IS A SINE WAVE?
HOW MUCH ATTENTION DID YOU PAY IN HIGH SCHOOL GEOMETRY?

THE THREE BASIC TRIGONOMETRIC FUNCTIONS

The three basic triginometric mathematical functions are SIN(e), COS(ine) and TAN(gent). They are defined as the ratios of various sides of a right triangle. A right triangle has one angle of 90o or Π∕2 radians. The sides of a right triangle are given names as shown below.

"Ø" is one Greek letter often used for angles. I don't know why and I really don't care. Where the name hypontenuse came from, I don't know or care either.

The RATIOS defining these functions are:-

RELATIONSHIP TO A CIRCLE

The angle Ø provides the relationship of these functions to a circle and cycles as well as to frequency and cycles per second or Hertz. Consider the following diagram.

If the line marked "C" (the hypotenuse) is rotated about the centre of the circle, since the diameter of the circle doesn't change, the hypotenuse will always be the same length. For convenience, this length has been chosen as 1 unit. When the angle of this line is zero, the opposite side will be zero and so both the tangent and sine will be the same. The cosine, or length of adjacent and hypontenuse will be 1 because both of these sides are the same when Ø = 0.

As the line rotates about the circle, the length of the opposite side, B, will get longer and longer until, at 90o, it will be the same length as C and the sine will be 1. Because the length of A will be 0, so will the cosine be 0. At 90o, the tangent will be infinite because of division by 0 ie. tan = opposite∕adjacent or 1∕0

If a plot of the length of the line B above is made, a sine curve results. In the diagrams below, the first plot represents 5 cycles and the second represents 20.

CALCULATING SIN

This site is about radio, not trigonometry. The above description is therefore only brief. There has been no indication yet of how the length of the side B is calculated. See the links below for more information. If you are unfamiliar with the calculus, the following formula looks complicated but it really isn't. SEE THE DESCRIPTION BELOW

This says the sum ("∑") of all values of the series from n=0 to ∞ of x raised to the power of (2n + 1) divided by (2n+1)! (a factorial or a! = 1 x 2 x 3 x 4 .... x n) alternating + and - gives the sin of x. If the first few terms of this series are shown for n = 0, 1, 2, 3 and 4, the formula becomes:-

This is a particular example of a power series called a Taylor series. It is also a particular case of a Taylor series called a Maclaurin series.

NOTE: X is a value in radians, not degrees. Radians is a measure of the length of the circumference of an arc compared to the radius of the circle rather than dividing a circle into an arbitrary number of degrees. Since the diameter of a circle = Πd (the number pi x diameter), there are 2Π radians in a circle. Π/2 radians is therefore 90o (90 degrees).

If the above series is calculated for Π/2 radians (90o) using only the terms shown we get:-

sin (1.5707963) = 1.5707963 - 0.6459641 + 0.0796926 - 0.0046818 + 0.0001604 = 1.0000034

Of course, we know the sin of Π/2 radians is 1 not 1.0000034 but if we continue the series forever we will get 1. This sounds a bit complicated for calculating such a simple value but it is the best way for all other values. Computers do it by using different convergant series depending on the size of the angle. For small angles sin(x) = x