Question of trig formatting Is there a difference between the following:
$$\sin^2x$$
$$x\sin^2$$
How about:
$$\sin(x)$$
$$\sin x$$
I'm new to trig and I've been confused on the formatting involved in trig, whether something is being multiplied or just applied to sin/cos/csc/etc. Can someone explain it to me?
 A: Yes, there is a difference between the first two: $\sin^2x$ means $(\sin x)^2$, while $x\sin^2$ is as meaningless as $x\sqrt{}$. The other two are a different story: $\sin(x)$ and $\sin x$ are simply two different ways of writing the same thing.
The thing to remember here is that $\sin$ (and $\cos$, $\tan$, etc.) are functions, not quantities. You apply them to some input: $\sin(x)$, for instance, is the number that you get when you apply the sine function to the number $x$. This is in keeping with the familiar $f(x)$ notation for functions. It’s traditional to omit the parentheses when the meaning is clear without them, mostly just to keep the expression from becoming too badly cluttered. Thus, we often write $\sin\pi$ instead of $\sin(\pi)$, and even $\sin 2\pi$ instead of $\sin(2\pi)$. When you see $\sin 2\pi$, be sure to think of it as $\sin(2\pi)$, not as $(\sin(2))\cdot\pi$. On the other hand, just about everyone would use parentheses in $\sin(2x+3)$, since $\sin 2x+3$ would normally be understood as $(\sin(2x))+3$, the result of applying the sine function to $2x$ and then adding $3$ to the result; $\sin(2x+3)$, on the other hand, is the result of applying the sine function to $2x+3$.
A: $\sin$ isn't a quantity, so $\sin^2$ means nothing. $\sin$ is a function, and $\sin x$ is just shorthand for $\sin (x)$. Normally, in function notation, you write the exponent after the function symbol, so $f(x)$ squared is $f^2(x)$.
A: $\sin$ is an operation, not a value. So for your first comparison:
$\sin^2x$ can not be written as $x\sin^2$  $\sin^2x$ is just an easier way to write $(\sin x)^2$
You are not multipling $x$ by $\sin^2$, you are squaring the value of $\sin$ of $x$. $\sin$ can never stand alone; it has to be used as the $\sin$ of something.

As for your second comparison, $\sin x$ is the same as $\sin(x)$.
You can usually exclude the parenthesis whenever there is no addition/subtraction. Although to make your math more readable, anything more than one variable/fraction should be wrapped in parenthesis.
A: I read somewhere (therefore it's true!!!!!  OK, tell me if you can cite something.) that Carl Gauss objected to the use of the notation $\sin^2 x$ to mean $(\sin x)^2$ on the grounds that $\sin^2 x$ ought to mean $\sin(\sin(x))$.
But language behaves the way it does and you can't push back the tide with a pitchfork, so $\sin^2 x$ in standard usage means $(\sin x)^2$.
