Question

What is the purpose of the cotangent, secant, and cosecant?

Answer 1

I can think right now, of two purposes.

Explanation:

Naming Ratios
For 3 numbers there are 6 ratios that can be made. Sine, Cosine and Tangent name 3 of the ratios, and the three you mentioned allow us to name all 6 ratios.

Avoiding division.

Example:
If we know that the angle of elevation to the top of a 12 meter tall building is `40^@` and we want to find the distance to the building, call it `x` we could use `12/x = tan40^@` so `x = 12/tan40^@` Now we get a decimal approximation for `tan40^@ ~~ 0.8391` and do the division.

OR use `x/12 = cot40^@` so `x = 12cot40^@ =`

We get `cot40^@ ~~ 1.1918` and multiply.

Also, in calculus many of us prefer to work with multiplication of functions rather than division and prefer, for example,

`y = x^3cotx` rather than `y = x^3/tanx`