What is the purpose, if any, of secant, cosecant, and cotangent? I’m an engineering student so I use sine, cosine, and tangent a lot, but almost never the other 3. Creating a new trig function just to be the reciprocal seems to be a bit redundant.
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They are simply mathematical tools to make things easier for us engineers to do complex calculations. Emergent properties from maths. Like complex numbers, you can’t physically realize the complex plane but in electrical engineering you can think of the imaginary part of calculations as the reactive power of a circuit (the energy released by capacitors and inductors) as a basic example.
All the trig functions are of the form: This is what I know, this is what I want to find out.
Most of the time, you know something about the hypotenuse of the triangle and the acute angle. The three functions you named are for when you know different things and are trying to work backwards. So for the tangent, you know the angle and get the ratio of the sides. For the cotangent you know the sides and work backwards to the angle.
On a polar plot, cosecant and secant give you horizontal and vertical lines respectively. These can be derived from the basic definition of vertical and horizontal lines, namely that x=c1 or y=c2 for some choice of constants. I’m not sure in how many scenarios you would need to plot straight lines on polar graphs, but the method is there.
It’s mostly just an archaic thing we don’t really need anymore.
Before modern calculators, the way you’d calculate sin, cos, and tan values was with a big table of values of the different trig functions for different angles.
In that context, it was useful to have separate columns for sec and cosec rather than everyone having to calculate that themselves.
Now, you’re probably not going to be doing trigonometry without a calculator, so it’s not really convenient to have special names for these values when it’s easy to just enter 1/sin(46) or whatever.
They’re mostly not worth teaching now, and to be fair any decent maths course won’t spend a lot of time on them. I suppose it’s useful to learn so that students don’t get too confused if they encounter it in older resources.
A triangle has **3** sides, say a, b, c. This means there are **6** ratios between two of them: a/b, b/a, b/c, c/b, c/a, a/c.
If we have a right triangle with hypotenuse c and the angle opposing a is φ, then those are in same order: tan(φ), cot(φ), cos(φ), sec(φ), csc(φ), sin(φ).
Or in names: tangent, cotangent, cosine, secant, cosecant, sine.
In other words, those 3 additional functions are simply the other ratios. It is ultimately a matter of taste which ones one prefers to use and which ones to forget. Historically, all six of them where used and thus named. But nowadays we skip half of them because they are only reciprocals of another one.
We use sin and cos instead of csc and sec because more often we are interested in a/c and b/c than the other way around. Furthermore, the sine and cosine theorems for _arbitrary_ triangles exist, which have no analogue with csc and sec. However, using tan instead of cot is completely random, there is really no good reason to prefer one over the other.
To express all six ratios in any triangle, two of them would suffice. Hence we one could argue that even tan has no merit.
It’s just a convention. You can work without them, but they have historical reasons for existing and some people are going to use them whether you like them or not, so you need to know them. There are also some situations when it’s much easier to use them than to fiddle around with fraction notation.
Also, if you want the reciprocal of sin(x), writing cosec(x) is much safer than using exponents to write (sin(x))^(-1), because that has the risk of being mixed up with sin^(-1)(x), which is the *inverse sin* of x, not the reciprocal of the sin of x.
“But we use sin^(2)(x) to mean the square of sin(x), not the sin of the sin of x, so why is sin^(-1)(x) different?” you might ask, and you might think that that’s terrible ambiguous notation, and you’d be right. Sometimes bad notation gets entrenched in the way people do maths, and there’s nothing you can do about it except strive for clarity yourself, and cosec(x) helps you avoid ambiguity.
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