3

I have seen $\sin^2 x$ in an equation and I don't understand how you can square $\sin$. So does $\sin$ squared $x$ equal $\sin x$ squared? Closed it has been answered

crvcio
  • 149

2 Answers2

6

Generally $\sin^{2}x$ is used as shorthand for $(\sin x)^{2} = (\sin x) \cdot (\sin x)$, which is different from $\sin x^{2}$. You can make it less ambiguous by writing the latter as $\sin(x^{2})$.

Edit: You just have to be careful when you see $\sin^{-1}x$, which does not mean $(\sin x)^{-1} = \frac{1}{\sin x}$, but rather the inverse sine, or $\arcsin x$.

DMcMor
  • 9,407
  • 1
    Thanks that has answered my question perfectly! – crvcio Nov 04 '20 at 19:26
  • 1
    It is best never to write $\sin x^2$ at all. Write either $(\sin x)^2$ or $\sin(x^2)$. Or, of course, the traditional $\sin^2 x$, as in the OP. – GEdgar Nov 04 '20 at 19:28
  • If you always use the functional notation $\sin(x)$ instead of the justaposition, the problem do not arise. – zwim Nov 04 '20 at 19:50
3

Carl Gauss (the most famous person to live on Earth in the 19th century, except for people who did not work in the physical and mathematical sciences) wrote (correctly in my view) that the notation $\sin^2 x$ ought to mean $\sin(\sin x)).$ But in his time it had already become conventional to use that notation to mean $(\sin x)^2,$ and that is what everybody learns in secondary school today and for many decades past.

The notation $\sin x^2$ usually means $\sin(x^2),$ and the latter is how I would normally write it.

Michael Hardy
  • 1
  • 4
    Trig power notation is one of the worst things to happen to basic function notation. So many students fundamentally do not understand inverse functions because of it. I don't know how many students I've tutored and had to explain it several times because of this discrepancy. I really wish we would collectively kick that nonsense notation to the curb and write what you mean. (I'd also opt for writing function notation explicitly everywhere, e.g. dropping $e^x$ notation in favor of $\exp(x)$, not writing $\sin x$ but rather $\sin(x)$, etc. but pipe dreams are exactly that.) – Cameron Williams Nov 04 '20 at 19:43