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In this question, I saw that there was no difference between $\sin^2(x)$ and $\sin(x)^2$.

However, I am curious what's the difference between $2\sin(x)$ and $\sin(2x)$?

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    $\begingroup$ $\sin 2x = 2 \sin x \cos x$ $\endgroup$ – Moo Mar 27 '16 at 5:23
  • $\begingroup$ Besides what Moo commented, try to see the difference when you take $\;x=\frac\pi2\;$ . $\endgroup$ – DonAntonio Mar 27 '16 at 5:24
  • $\begingroup$ Qualitatively, $\sin 2x$ has amplitude $1$ and period $\pi$, whereas $2\sin x$ has amplitude $2$ and period $2\pi$. $\endgroup$ – Bungo Mar 27 '16 at 5:27
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    $\begingroup$ There is no difference between $\sin^2(x)$ and $\sin(x)^2$ simply because we define $\sin^2(x)$ as what you get squaring $\sin(x)$; this is not an identity or anything deep, but notation. $\endgroup$ – pjs36 Mar 27 '16 at 5:31
  • $\begingroup$ $\sin^2(x)$ and $\sin(x)^2$ are analogous to $(\sin\cdot2)(x)$ and $\sin(x)\cdot2$, whereas $2\sin(x)$ and $\sin(2x)$ are probably analogous to $\sin(x)^2$ and $\sin(x^2)$. $\endgroup$ – L. F. Mar 16 '19 at 3:12
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I think that aside from that fact that $$\sin 2x = 2\sin x\cos x,$$ the difference should be patently clear once you have the graphs. enter image description here

Notice that $2\sin x$ means take the value $\sin x$ and double it. Where as $\sin 2x$ means take the value $x$, double it, and then apply the $\sin $ function to that value.

Also, I do not believe it is universally accepted that $$\sin^2 (x) = \sin (x)^2,$$ since $\sin (x)^2$ can be interpreted as $$\sin ((x)^2).$$ For a less ambiguous case, take for example $\sin (2xy)^2$ is can easily be interpreted as $$\sin 4x^2y^2$$ or $$\sin^2 (2xy).$$ So I believe there is a difference.

For the sake of clarity, $\sin (x)^2$ should really be written as $$(\sin x)^2$$ if you mean to convey $\sin^2 x$.

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  • $\begingroup$ What program are you using to graph? $\endgroup$ – Michael McQuade Mar 27 '16 at 5:44
  • $\begingroup$ Desmos online graphing calculator. $\endgroup$ – Em. Mar 27 '16 at 5:45
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$2\sin(x)$ is multiplying the $\sin$ function by 2. Whereas the $\sin(2x)$ is inputting a value of twice $x$ into the function. Note that $\sin(2x) = 2\sin(x)\cos(x) \neq 2\sin(x)$

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$2\sin x$ means first you take the sine of $x$. Then you multiply the result by 2.

$\sin 2x$ means first you multiply $x$ by 2. Then you take the sine of the result.

$\sin 2x$ can, and sometimes should, also be written as $\sin(2x)$.

It is not ok to pull constant multiples out of trig functions.

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Really, $\sin(x) ^2 = \sin^2(x)$ only because we adopted the function notation used prior, namely $f(x) ^2 = f^2(x)$. In most cases I still use the former to avoid confusing differentiation or iteration, but trig functions still use the latter. Now, this notation only holds for powers... When you say $2\sin(x)$ you are making a sine wave with an amplitude of $2$. When you say $\sin(2x)$ you are talking about a sine wave with twice the frequency of the standard sine wave.

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$\sin(2x)=2\sin(x)\cos(x)$ its double angle formula

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