Triangles have extraordinary properties. By “extraordinary” I don’t mean anything mystical like pyramid power (remember that?) or the Bermuda Triangle. Rather, I mean something far more interesting, if less exciting to the superficial observer.

Trigonometry (literally “triangle measuring”) allows us to explore some of those properties. The properties we’ll be playing with are derived from ratios between the sides of a right-angle triangle, but it’s important to note that trigonometry is applicable to all triangles. The central idea concerns the ratios between the length of a triangle’s sides and its angles.

Figure 1 shows an arbitrary right-angled triangle, drawn between the points A, B and C. By convention, the associated angles are called A, B and C (i.e. with capital letters) and the sides opposite each point are called a, b and c respectively. (It’s also common practice to use the Greek alphabet for naming angles, but I’ll avoid that wherever possible.) In a right-angled triangle such as ABC, the sides also have special names: the hypotenuse is opposite the right angle, while the opposite and adjacent are, clearly enough, opposite and adjacent to the angle you’re playing with.

Those strange beasties: sine, cosine, tangent

Actually the ratios, despite their odd names, are actually quite simple. It’s just that the names don’t suggest their meaning in any way.

Sine

The sine (abbreviated to sin) is the ratio between the opposite and hypotenuse, which is to say opposite divided by hypotenuse or a/c. To put it more succinctly:

sin(A) = opposite/hypotenuse = a/c

Cosine

The cosine (cos) is the ratio between the adjacent and the hypotenuse:

cos(A) = adjacent/hypotenuse = b/c

Tangent

Not to be confused with a line just touching a circle, the tangent (tan) is the remaining ratio, between the opposite and adjacent:

tan(A) = adjacent/hypotenuse = b/a

(Actually there is a good reason why we use the word tangent, which I’ll get to presently.)

Using the ratios for measurement

Let’s suppose we have a triangle, necessarily right-angled but otherwise arbitrary, in which we know the length of one side and one angle. Using trigonometry we can calculate the lengths of the other sides.

Since sin(A) = sin(50°) = a/c, it follows that

a = c x sin(50°) = 12 x 0.766 = 9.192

Magic! This is in fact how we determine the height of mountains, for example. Since it’s relatively easy to measure the distance between the mountain’s base and where you’re standing to observe it, all you need is to measure the angle between the ground and the mountain’s peak. Then the calculation is just as I described it above.