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So with $y = \sin (4x)$ I divide $4$ by $2\pi$. All youtube tutorials show $-1$ is obtained by doing $$\sin(3 \pi /8)$$

I've tried this in a calculator with the mode set to radians, but no matter what variation, I can not get a negative value to appear.

What $x$ will get to $-1$?

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Since $\sin\left(\frac{3\pi}{2}\right) = -1$, then $4x = \frac{3\pi}{2} \implies x = \frac{3\pi}{8}$ will work to have $y = \sin(4x) = -1$.

The problem with checking $\sin\left(\frac{3\pi}{8}\right) = \sqrt{\frac{1}{2}\left(1 + \frac{1}{\sqrt{2}}\right)} \approx 0.924$ is that it's $x = \frac{3\pi}{8}$, with $4x = \frac{3\pi}{2}$ being the argument to $\sin$, to get a result of $\sin(4x) = -1$.

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  • $\begingroup$ Ohhhh haha, that's great. Thank you so much. I am so crap at this, so if you can please help with one last part. Why does the sin4x = 2pi / 4 x 4 (this part I understand) AND THEN we still use the 4 inside the sin function? I thought it would have been used up to make 3pi/8? $\endgroup$ – Anon Jan 24 at 4:58
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    $\begingroup$ @Anon You're welcome. I don't understand what you mean by $\sin 4x = 2\pi / 4 x 4$. Note $\sin(4x)$ means to take any $x$, multiply it by $4$ & then apply the $\sin$ function to it. As for using the $4$ inside the $\sin$ function, if it helps any, consider it to be $y = 4x$ and $\sin(y)$ instead. As for $4$ being "used up to make 3pi/8", the key element is that for the result of $\sin$ to be $-1$, the input needs to be $\frac{3\pi}{2} + 2k\pi$, but we use $k = 0$ here. As the input is $4x$, for it to be $\frac{3\pi}{2}$ means $x = \frac{1}{4}\left(\frac{3\pi}{2}\right) = \frac{3\pi}{8}$. $\endgroup$ – John Omielan Jan 24 at 5:08
  • $\begingroup$ You answered it perfectly. Thank you so much! $\endgroup$ – Anon Jan 24 at 7:01

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