# Conventions for function notation

I'm in year 11 right now and I just had a brief discussion with my maths teacher about function notation in trigonometry.

For a test, I wrote this,

sin(50)^2


I assumed that would be interpreted as sin(50)*sin(50)

But I was told the correct notation for this is is

sin^2 (50)


or optionally

(sin (50))^2


I'm curious if that is the 'proper' mathematical convention, or just how things are taught in high schools?

I'm Australian, in case there are some regional differences.

Thanks for the help!

• I don't remember where I read it, but it seems Gauss strongly rejected the $\sin^2(x)$ notation, because it should mean $\sin(\sin(x))$ and that only, so you're not in bad company if you don't like it. -- By the way, tell your teacher to type "sin(1/2)^2" and "sin^2(1/2)" in google and check what comes up... – Myself Mar 2 '11 at 0:45

This is a weird notational bug specific to trigonometric functions; chalk it up to historical inertia. We write $\sin^2 x$ for $(\sin x)^2$ but for a generic function $f$, more often than not $f^2(x)$ means $f(f(x))$ and does not mean $(f(x))^2$ (or $f(x^2)$). On the other hand, $\sin^{-1} x$ means $\arcsin x$ rather than $\csc x$...
It is preferable to include extra parentheses when in doubt. Generally I would interpret $f(x)^2$ as $(f(x))^2$ but it is less clear whether $f(\log x)^2$ means $f((\log x)^2)$ or $(f(\log x))^2$.