Trigonometry and Complex Numbers

John Baez

December 6, 2011

When I teach calculus I keep running into students who regard trig identities and trigonometry in general as something that requires feats of memorization. Personally, I think the best way to really understand this stuff is to understand complex numbers.

To "understand complex numbers", you first need to understand the real numbers in a geometrical way, in terms of a line:

1. Understand how addition of real numbers is related to translating (sliding) a line: adding x to the coordinate of a point on the line translates the point x units to the right.

2. Understand how multiplication of real numbers is related to dilating (stretching or squashing) a line: multiplying the coordinate of a point on the line by x dilates it by a factor of x.

3. Also, understand how multiplication by -1 is related to reflecting a line, which is a rather nonobvious special case of item 2.

1. Notice that x² = -1 has no solution in the real numbers because you can't dilate by a factor of x twice and have the result be a reflection, no matter what x is. (In the process, learn why the product of two negative numbers is positive: sadly, many students are left with the impression that it's only true "because the teacher says so", which makes it impossible for them to understand complex numbers.)

2. Notice that x² = -1 does have a solution if we go up to the plane: x can be a quarter-turn to the left or to the right! We use the name i for the number that gives a quarter-turn to the left.

3. More generally, understand how multiplication of complex numbers is related to dilating and rotating of the plane.

4. Also, understand how addition of complex numbers is related to translating of the plane.

5. Understand that to make addition simple, we want to use Cartesian coordinates and write our complex numbers as x+iy, while to make multiplication simple, we want to use polar coordinates and write them as r exp(i θ).

   Here exp(i θ) has no meaning other than "the number by which you multiply to rotate by an an angle θ counterclockwise"! Or in other words: "the number on the unit circle which is an angle θ counterclockwise from the number 1." You don't need to know anything about exponentials for this — in fact, it's best if you don't! (So don't bring in the power series for the exponential at this stage; it doesn't help.)

   Since we have these two ways to write the same number, we have

   x + i y = r exp(i θ)

6. From item 5, you can see that

   exp(i θ) exp(i θ') = exp(i(θ + θ'))

   since it merely says that rotating by the angle θ and then the angle θ' is the same as rotating by the angle θ+θ'.

7. For points on the unit circle r = 1, so by item 5 we have

   x + i y = exp(i θ)

   So, for points on the unit circle x and y are functions of θ. We make up names for these functions:

   x = cos(θ)

   y = sin(θ)

   Again, you don't need to know anything about trig for this; this is the definition of sine and cosine. So, by definition we have

   cos(θ) + i sin(θ) = exp(i θ)

8. The Pythagorean theorem says that

   x² + y² = r²

   For points on the unit circle, r = 1, so item 7 implies

   cos(θ)² + sin(θ)² = 1

   This is the most important trig identity.

9. Understand the formula for multiplication in Cartesian coordinates:

   (x + iy)(x' + iy') = (xx' - yy') + i(xy' + x'y)

   It's hard to see this directly in a geometrical way, since Cartesian coordinates are not suited to describing rotations/dilations. But it's easy to see algebraically once you know i² = -1 and the distributive law, and both these facts have good geometrical explanations (see items 3-5 for this part of the story).

10. Use a little algebra to conclude that

    cos(θ) = (exp(i θ) + exp(-i θ))/2

    sin(θ) = (exp(i θ) - exp(-i θ))/2i

    It's good to understand these using pictures, too.

11. Finally, see what happens when you multiply a sine and a cosine. By items 6 and 10 it's obvious that you will be adding and subtracting angles! This fact is much more important than the actual formulas:

    cosA cosB = [cos(A+B) + cos(A-B)]/2

    sinA sinB = [cos(A-B) - cos(A+B)]/2

    sinA cosB = [sin(A+B) + sin(A-B)]/2

    which nobody should bother to remember, since they're easy to derive when needed, by precisely the calculation sketched here: turn the sines and cosines into exponentials, which are easy to multiply; then multiply them; then turn them back.

By the way, I think that any course that covers trig identities like the ones we're talking about here without introducing complex numbers is being a bit silly. z = exp(i θ) is just a name for a point on the unit circle; it's like tying one hand behind your back to never call this point by its true name, forcing yourself to work only with its coordinates cos(θ) and sin(θ).

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