Sine Function Explained in 5 Minutes

Sine Function Explained in 5 Minutes

The sine function comes from trigonometry - the study of angles. For this reason it's called a trigonometric function, but its importance goes beyond trigonometry. It's a key function in mathematics and is used extensively in physics.

If q is an angle, then the sine function is denoted by sin(q). So what is it?

Draw a circle of radius 1. Now pick any point on the circle and draw the radius from the point to the center. This line makes an angle with the horizontal, so let's call this angle q. Then sin(q)=v where v is the vertical distance of the point above the horizontal.

Of course, this definition immediately suggest another function f(q)=h, where h is the horizontal distance of the point from the vertical. As we might expect, this function is closely related to the sine function and is known as the cosine function cos(q).

When sine is used in other areas of math and physics it's denoted by general function notation f(x)= sin(x) where x is a variable that can assume any value.

As with many mathematical functions, sin(x) can be written as infinite series which seem to bear no relationship to the above definition!

sin(x)=x-(x^3/3!)+(x^5/5!)-(x^7/7!)+(x^9/9!)...

where n! is the factorial function n!=1*2*3*4*.....*n

Visually sin(x) is impressive, because if you plot it on a graph you get a beautiful wave! This explains why it is so useful in physics, where it appears in the description of all sorts of wave phenomenon.

It also has another feature that's not often mentioned - it satisfies a recurrence relation..

sin(n*x)=2*cos(x)*sin((n-1)*x)-sin((n-2)*x)

where n is an integer n=2,3,4,...

Recurrence relations are interesting and powerful things because they let you define a function in terms of itself! To see how this works consider the above recurrence relation and let x=a, where a is a constant, so it's no longer a variable. Then we get..

sin(n*a)=k*sin((n-1)*a)-sin((n-2)*a) where k=2*cos(a) is a constant.

Notice that the only variable in this equation is n, so define the function f(n)=sin(n*a) and we get..

f(n)=k*f(n-1)-f(n-2)

So if a, f(0) and f(1) are given then f(n) can be calculated for any n. It defines itself!

Sure enough, if you plot f(n) for n=2,3,4,5,... you get the beautiful sine wave, and the three numbers a, f(0) and f(1) determine the amplitude, wavelength and phase of the wave. Nature uses recurrence relations to generate all sort of amazing things, which is why the sine function shows up so often when we try to describe these things in physics. But we could also bypass the sine function and describe them with a recurrence relation!

Content written and posted by Ken Abbott abbottsystems@gmail.com