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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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.
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