Sine of the times.

[Nigel (UK)]

(06-12-2024 07:32 PM)Matt Agajanian Wrote:  Hi all.

Just befuddled. How did sin(x)² (cosh(x)², csc(x)², etc.) come to be interpreted as sin² x or (sin x)², etc.? To me sin(x)² represents x x² SIN. But, clearly, I must have missed the memo.

The problem with \(\sin(x)^2\) is that it is ambiguous - is it just \(x\) that's being squared, or is it \(\sin(x)\)? Writing \(\sin^2x\) makes the intent plain without the need for (extra) brackets. Since squared trig functions are so common, it's a very popular time-saving notation.

You did indeed miss the memo. I've found a reference (scroll down the page to find it) to Laplace having used this notation; Gauss found this odious, because he felt that \(\sin^2(x)\) should mean \(\sin(\sin(x))\) (and he has a point!) but since the repeated sine is used so rarely this isn't an issue in practice.

Nigel (UK)

(Added) Just to confuse things, \(\sin^{-1}x\) means the inverse sine of \(x\), not \(1/\sin(x)\)!