Wednesday, February 10, 2010

Wednesday Math. Vol. 108: Trigonometry and Duality

The word opposite isn't used much in math, because there are so many different ways to be opposite. Some elementary algebra books will say that 3 and -3 are opposites, but soon enough they stop using that word and start calling them the negatives of one another. You could also say that 3 and 1/3 are opposites, but the common word used is reciprocal. You could say that squaring a number and taking its square root are opposite actions, but the word used there is that they are inverse actions of one another. The list goes on and on.

One common usage of "opposite" in math deals with the geometry of a triangle. Pick any angle, and two of the sides help make that angle, so those sides are adjacent. The other side is opposite, so any angle in a triangle has exactly one opposite side and vice versa, every side has an opposite angle.

When teaching trigonometry, there is a commonly used mnemonic for the formulas for sine, cosine and tangent,
soh cah toa. Teachers tell students to pretend it's a Native American name. It stands for the following three equations.

Sine A = opposite/hypotenuse
Cosine A = adjacent/hypotenuse
Tangent A = opposite/adjacent

There are three more basic trig functions, and they are defined as reciprocals of the first three.

Cosecant A = 1/Sine A
Secant A = 1/Cosine A
Cotangent A = 1/Tangent A


So far, so good, but let's take a hackneyed metaphor and turn it into a literal thing. Let's look at it from a different angle. If we take Angle B as our point of reference, the hypotenuse doesn't change, but what we call the adjacent and the opposite switch. Because this is a right triangle, the two small angles A and B must add up to 90°, since A + B + 90° = 180°. Any time two angles sum to 90°, they are called complementary, so A is complementary to B and B is complementary to A. That's where the "co" in cosine, cotangent and cosecant come from.

Sine A = Cosine B
Cosine A = Sine B
Tangent A = Cotangent B
Cotangent A = Tangent B
Secant A = Cosecant B
Cosecant A = Secant B

In equations, the six trig functions are shortened to sin, cos, tan, cot, sec and csc. The idea that we can interchange every function with its complementary function means we have a system with duality, two ways of looking at any equation. For example, there is a trig identity involving tangent and secant.

tan² A + 1 = sec² A

This means there must be a dual identity with cotangent and cosecant.

cot² B + 1 = csc² B

The letters A and B are completely arbitrary, so by relabeling, we can also say

cot² A + 1 = csc² A

Trig identities can get messier, but if we are dealing with a single angle, every identity has its dual, two true statements for the price of one. Take this identity.

sec A = cos A + sin A × tan A

This means its dual must also be an identity.

csc A = sin A + cos A × cot A

Not every system in math has duality, but those that do are very powerful and it makes proving theorems easier. Next week, we'll look at another dual system even more powerful and fundamental than trigonometry, logic.

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