Anyone who knows trigonometry knows $\sec(x)=\frac{1}{\cos(x)}, \csc(x)=\frac{1}{\sin(x)}, \cot(x)=\frac{1}{\tan(x)}$.
Writing this out in words, we get the following:
- Secant of x = (1 / COsine of x)
- COsecant of x = (1 /Sine of x)
- COtangent of x = (1 / Tangent of x)
Here's my question: where do these naming COnventions come from (ignore the pun), what's their history? And much more importantly, why are the relations (in terms of their names) the way they currently are. Wouldn't it make more sense for $\sin(x)$ to be related to $\sec(x)$, and $\cos(x)$ to be related to $\csc(x)$? Or even a more extreme example, like $\sin(x)$ being the reciprocal of $\cos(x)$, $\sec(x)$ being the reciprocal of $\csc(x)$, and $\tan(x)$ being the reciprocal of $\cot(x)$?
And lastly, why were the terms sine, secant, and tangent chosen for their respective meanings?
If you're able to help out, thank you so much in advance!