Trigonometry by triangles

This week, we’re going to be talking about trigonometry. Now, I must confess that I’ve often heard people say that the worst course they ever took in high school or the one they liked the least was trigonometry. And sometimes they will also mention algebra. Sorry, Alberto. Now trigonometry has a bad press. And I think I know why. I will point it out when we get there what I think the reason is. But it’s such a beautiful subject. And it’s so ancient. It contains some of the oldest mathematical concepts that ever existed. It certainly goes back to at least the ancient Egyptians. I like to think that trigonometry was invented the day somebody wanted to measure something.

For example, the height of a very high cliff. Let’s call that height capital A. And we don’t know how to measure it in practical terms. Because we can’t just go over there and do it directly. And someone one day had this very, very bright idea. There was a tree not far from them. And they could measure the height of the tree directly. It was lowercase a, little a. And then they imagined a straight line going from the top of the cliff to the top of the tree and hitting the ground, let’s say, where they’re standing. Perhaps they even had a ray of sunshine that helped them imagine that straight line.

In any case, once you had that straight line, you see that you have two triangles in your picture. Now, you can measure the base of the smaller triangle. That’s small b, because it’s just the length on the ground between you and the base of the tree. And similarly, you can measure capital B, the distance to the base of the cliff. Now, once you have these measurements, you use the fact that your triangles are similar. And you deduce from that that capital A over B is equal to small a over b. That enables you to solve the equation for capital A in terms of known quantities. You have found capital A. How clever that is?

Now, someone had an even brighter idea later, maybe even the same person. They realized that, in this whole discussion, what really matters is not these three numbers separately, but the quotient a over b, which depends only on the angle theta. I mean by theta, the angle that is formed at the point where you’re standing by the triangles. Now once you make this realization, you then realize that you don’t need a tree. In fact, all that counts is theta. You could set up in your living room some triangle that gives you the angle theta on the left as indicated. You have to make sure it’s a right-angled triangle. And then you’d measure a and you’d measure b.

And you’d find a over b. And of course, you should record it for later use in case you need it again later on. And so you have to give it a name. And you call it the tangent of theta. So here’s an important definition. The tangent of an angle theta is defined by constructing a triangle as you see and defining the tangent of theta to be the opposite over the adjacent length. That’s a over b. Opposite as in the sense of opposite the angle. And adjacent, of course, means it’s next to the angle.

Similarly, if you call the length h that corresponds to the hypotenuse of this right-angled triangle, then the sine of theta is defined to be opposite over hypotenuse, a over h. And the cosine is defined to be adjacent over hypotenuse, b over h. We have now defined the three most important trigonometric functions in terms of triangles. Let’s talk about how we measure angles now that we’ve seen how important they are. Probably the first way that you ever measured angles like me was in terms of degrees. That is, if you start off in the standard position that I’ve indicated here and you do one full revolution, that is, by definition, 360 degrees.

We owe this to the Babylonians who thought that 60 was a great number to base their arithmetic on. And they were right, because 60 is divisible by so many things. However, 360 then for a full revolution. So a quarter revolution, for example, corresponds to 90 degrees, what we call a right angle. And it is a classical result in Euclidean geometry, which by the way would be a subject well worth a course just on its own, that the sum of the three angles of a triangle equals 180 degrees. Now as you’ve probably noticed, math nerds like me, we love Greek letters. One alphabet is not enough. And so the three angles in this triangle are called alpha and beta and gamma.

And the theorem says that alpha plus beta plus gamma in degrees is equal to 180. These facts allow you to calculate the tangent and the sine and so forth of certain angles without actually doing any measurement. For example, suppose your angle theta is 45 degrees. Because the sum of the three angles is 180, you can easily calculate from that that your top angle is also 45 degrees. It follows that your triangle is an isosceles triangle. Those two sides are equal. Let’s give them unit length. Then by the theorem of Pythagoras, the hypotenuse has length square root of 2. Once you know that about the triangle, you can just read off what the tangent of 45 degrees is, for example.

It’s the opposite over the adjacent, 1 over 1, which is 1, and so forth for the sine and the cosine. Now, 30 degrees is a little trickier but still possible. It turns out that the top angle then is worth 60 degrees. And you can prove geometrically that the vertical height of that triangle is a half of the hypotenuse. So giving them lengths 1 and 2, we then deduce from Pythagoras that the other length of the triangle is square root of 3. Once we have all that, we can read off, for example, the tangent of 30 degrees, opposite over adjacent, 1 over square root of 3 and so forth for the other two functions.

Now, most math students who have done trigonometry are expected to know these values of sine, cosine, and tangent. You can deduce certain others from them, the complementary formulas. Once you know the tangent of 30, then the tangent of 60 is the reciprocal of that. And the sine of 30 is the same as the cosine of 60. You can see that right from the definitions. And cosine of 30 is the sine of 60 and so forth. Now for hundreds of years, people have been solving triangles using partial information on a triangle in order to get other information about the triangle. Here’s a typical example. You have a triangle PQR. You know that at P the angle is 30 degrees.

And you know that the length PQ is 8. Problem? Calculate PR. Well, we know from definition of cosine that the cosine of 30 degrees is PR over PQ, adjacent over hypotenuse. We also know the cosine of 30 degrees. It’s root 3 over 2. Therefore, we deduce that PR is 4 root 3. By the way, the length PR is called in the jargon the projection of PQ onto the horizontal axis. Let’s look at a couple of other classical results involving angles in Euclidean geometry.

If you want to know the area of a triangle and you have two of the lengths and you also know the angle that is formed by those two lengths theta, then it turns out that the area of that triangle is 1/2 AB sine theta. A beautiful theorem. Another theorem, which is very famous in geometry, is called the law of sines. If you know the three angles A, B, and C and if you know the lengths of the sides opposite those three angles, then it turns out that the quotient of each sine divided by its opposite length is the same for all three angles. This enables you to, so-called, solve a triangle.

For example, if you know two sides of the triangle and one angle or if you know two angles and one side, you can figure out the others. People have been doing that for a long time. People who are geometers, of course, but also surveyors or engineers or architects. And it’s a little bit of a forgotten science to tell you the truth. Somewhat sad that all that beautiful Euclidean geometry is not really being studied very much anymore. Why not? Well, because there’s so much mathematics and so little time. And you have to do what counts the most. And that’s what we’ll do when we start up again later on.