Understanding Trigonometry: A Matter of Complex Circles

When students are first introduced to trigonometry, they are often shown a diagram like this one as a way of explaining the sine and cosine functions:

See, theta is an angle of a right triangle, and if the hypotenuse equals one then cosine and sine give us the lengths of the adjacent and opposite legs, respectively. As theta changes, the triangle changes, and sine and cosine return different values. All of this is correct and pretty easy to grasp. The problem is, as a conceptual aid, it starts crumbling the moment theta exceeds 90 degrees.

I think a better way to come at this is to recognize cosine and sine as the coordinates of a point that is traveling around the circumference of a unit circle. Round and round it goes, non-stop forever, and no matter where it is, the number of radians traveled translate directly into a pair of X and Y coordinates, thanks to those lovely functions.

Now, if you can get in the habit of thinking about sine and cosine that way, as the X and Y coordinates along the circumference of a circle, I think a lot of related trigonometric ideas start making better intuitive sense. For example, why does a sine wave repeat? Because a point is traveling around a circle, repeating its path indefinitely. Personally, I find that I grasp an explanation like that one a little more quickly than if I try to relate sine waves to the triangle that stared this post.

And the triangle construction fits right in. If you need to look at something like a triangle, then draw a radius from the origin to the point and then drop some perpendiculars. There’s your triangle, just like before.

Now, armed with our unit circle understanding of sine and cosine, I want to take a detour into algebra and see what light we can shed on a certain famous identity. X and Y coordinates give us a grid that is similar to the complex plane. I hope you are already familiar with complex numbers. The idea is, the square root of -1 cannot be any number on the real number line, so we arbitrarily declare the square root of -1 equal to an “imaginary” number that we call i. This is useful; the square root of -9 is then 3i, because 3i*3i = (i*i)*(3*3) = -1*9 = -9, and so too lots of other numbers. Numbers built with i cannot be on the number line, and so we go two-dimensional and posit what we call the complex plane. It looks like this:

The complex plane looks and operates in a manner that somewhat resembles Cartesian coordinates. Sine and cosine– which we think of as X and Y coordinates– can be adapted to the complex plane, and this is done by means of a famously powerful and elegant relationship called the Euler Equation:

What has happened is that cosine and sine have been combined into one complex number. Cosine is the real part while sine is the imaginary part, and this is unsurprising if we already have the understanding that cosine returns an X coordinate and sine returns a Y coordinate. The Euler Equation takes us one step further and not only converts sine and cosine into the equivalent complex number, but it also shows that this complex number happens to equal the constant e, raised to a power of i.

Cosine and sine, we must be careful to remember, describe coordinates of a point on the circumference of a circle. No matter what the values of theta, sine and cosine never leave the circle. Consequently, e^iθ never leaves a circle either. Whatever e^iθ is, it is always on the circumference of some circle, its location on the circumference specified by theta. Round and round e^iθ goes.

Why the Euler Equation is true, I don’t want to get into here. The proof involves Taylor Series, and while it is not all that difficult– rather fun, actually– it strays too far from what I want to focus on in this post. If you need to research it yourself, I don’t expect it will give you much trouble. On the other hand, if you are willing to accept the equality as given, I have one last thing to show you.

Now, if the starting point on our unit circle is at X=1, Y=0, then where does a theta of 180 degrees (that is to say, in radians, θ=π) put us? If you’ve understood everything to this point, and if you are thinking of sines and cosines as returning coordinates for points on a circle’s circumference, then you can probably see that 180 degrees would put us at X=-1, Y=0. Or, in complex numbers, 0i-1. From this, we have the famous Euler identity:

Rather obvious, really.

Obvious, but if we had tried to get here by thinking of sine and cosine in terms of right triangles, then the Euler Equation would have been bewildering. Many people today still regard the Euler Identity as something quasi-magical. It’s beautiful, but it’s not magical. A reinterpretation of sine and cosine makes the identity easy.