Sines of the Times

Trigonometry is a mystery to a lot of people. First off, what is it in the first place, and second, what is it for?

First answer first: It’s triangles. It’s about how the sides join together and the way this affects the rest of the triangle.

Imagine a ladder leaning against a wall. How do you find the best way to lean it? If the first rung is too far away from the wall, the ladder will slip when you climb up. If the first rung is too near the wall, it will not slip, but the ladder will be harder to climb up. The best way seems to be a compromise. The “leaning factor” needs to be optimised. This “leaning factor” is a measure of how steep (or not) the ladder needs to be to be safe to climb up. This measure has a unit, and that unit is “degrees”.

For the ladder to be stable, it needs to be at about 60 degrees with respect to the ground (about 2/3 of a “square” lean (that is, if the ladder is vertical))

The picture that the ladder makes is called the “angle” it makes with the ground.

Trigonometry is just the study of relationships between “angles”. Questions have arisen, such as this: If my ladder is five meters long, how far up the wall will I be able to climb using the “optimised leaning factor”? Will I be able to clean that window which I know is exactly four meters above the ground?

The answer is “Yes” because the answer is that the top of the ladder will be at 4.33m up the wall.

Here’s the picture of what is happening:






The ratio of the length of the ladder to the distance up the wall is that same for all angles of “2/3 of a “square” lean”. In fact, these ratios are all the same for each “lean” you care to name.

They have names, too.

Here is a typical triangle they relate to:





The three sides are named (O)pposite, (A)djacent and (H)ypotenuse, and the ratios go like this:

Sine = Opposite/Hypotenuse

Cosine = Adjacent/Hypotenuse

Tangent = Opposite/Adjacent,

Or, as one of my students taught me, SOHCAHTOA.

But why bother? The answer is that the numbers have already been worked out, so if you do not know the angle but know the sides, you can look it up on the calculator. Also, if you know the angle and the length of one side of the triangle, you can find the length of the other sides as well.

Oh, one final thing, it works best if the triangle has one angle which is the same as the ladder standing up vertically.

That kind of “angle” is called a “right angle”, to give the right angle on things. Of course, the boffins figured out how to do all this even when the angle is not “right”, but that is not the point, not to put a too acute point on it.

Suffice it to say, sines, etc, go far beyond mere triangles, and in fact have more applications in math than you can shake a proverbial SOHCAHTOA at, but that would be a sign of going off on a tangent.