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I am wondering if it is possible to solve the equation \begin{equation} \sin(x) = 0.4. \end{equation}

If it is possible to solve this, how does one do so?

yoknapatawpha
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Ryuzaki
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    $\sin x=0.4$ is possible, but $\sin x=0,4$ is not because of the range of the $\sin$ function is $[-1,1]$. –  Feb 02 '14 at 21:05
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    okay, then this is based on the range of sin function which means that sin x =4/3 is not possible, is that right? – Ryuzaki Feb 02 '14 at 21:08
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    Yes, exactly. Perhaps you are asking of the inverse sine function, $\arcsin$? –  Feb 02 '14 at 21:36

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Yes you can, if you plot the function $\sin x$ then you can see that what you want are the values of $x$ where $\sin x$ is 0.4. This is shown by the red line in the plot below.

enter image description here

First thing to note is that because $\sin$ is a periodic function, there is more than one solution, as witnessed by the fact that the red line hits the black line in more than one place. You usually choose the range that you want $x$ to be before you start, in this case, for $\sin$, between $-90$ and $90$ degrees is usual.

Second thing to note is that if you'd chosen a value for $\sin(x)$ outside the range $-1$ to $1$ then there's no way the red line could hit the black sin wave, and so there would be no solution.

Also, technically, to solve an equation like this we apply the inverse $\sin$ function, $\sin^{-1}$ (a.k.a. arcsin), to each side of the equation. Applying an inverse function to itself cancels out, so we have

$$\sin(x) = 0.4$$ $$\sin^{-1}(\sin x) = \sin^{-1}(0.4)$$ $$ x = \sin^{-1}(0.4)$$

you can look up the result on a calculator. (Apart from a few special values like 0 degrees, 30 degrees, 45 degrees, 60 degrees, 90 degrees and so on, the values of $\sin^-1$ aren't easily known and so the calculator has to follow a numerical algorithm to get you the answer.)

The answer is 23.6 degrees, to 1 decimal place, as shown by the green line.

TooTone
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    You may be interested to know that the only rational values assumed by $\sin x$, where $x$ is any rational number of degrees, are the numbers $0,\pm\frac12,$ and $\pm 1$. (This is usually stated when the argument is measured in radians rather than degrees, so we would normally speak of restricting $x$ to being a rational multiple of $\pi$, but this is equivalent.) – MPW Feb 02 '14 at 22:07
  • @MPW that's a good point. I was also thinking that the value of $\sin(x)$ for, say, $x=45$ degrees is known precisely but this isn't a rational number. – TooTone Feb 02 '14 at 22:30