I typed the derivative of $\sin(x)$ at $x=0$ into my calculator, and it outputted $0.0174...$ I was expecting the answer $1$. What went wrong?
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Tian Vlasic
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no1dea
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1Put your calculator in radians mode. The derivative of $\sin x$ is $\cos x$ only if $x$ is measured in radians. This might be relevant. – kipf Jun 25 '23 at 12:44
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@kipf ♂️ Of course, thanks! – no1dea Jun 25 '23 at 12:53
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@JohnDouma: That is true, but the derivative of $\sin(x^\circ)$ is not $\cos(x^\circ)$. Since angles are a dimensionless quantity, the degrees symbol is actually a shorthand for the constant $\frac{2\pi}{360}$, and so by the chain rule, the derivative of $\sin(x^\circ)$ is $\frac{2\pi}{360}\cos(x^\circ)$. – Joe Jun 25 '23 at 12:56
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You also missed a $0$ when you typed up the answer you are getting – Ross Millikan Jun 25 '23 at 13:04
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@RossMillikan: Sorry, that was my mistake. I edited the question to transcribe what the calculator was saying, since I thought it would make it more accessible. – Joe Jun 25 '23 at 13:05
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Your calculator is in degrees mode. As such, it is calculating the derivative of the function $\sin (x^\circ)$, which is $\dfrac{2\pi}{360}\cos(x^\circ)$. At $x=0$, we have $\dfrac{2\pi}{360}\cos(x^\circ)=\dfrac{2\pi}{360}\cos 0=\dfrac{2\pi}{360}$, which agrees with the result given on the calculator.
By contrast, the derivative of the function $\sin x$ is $\cos x$, and $\cos0=1$. Therefore, if your calculator is set to radians mode, it ought to output $1$.
Joe
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@no1dea: No problem! When I was calculus student, I ended up getting so tired of remembering whether to use degrees/radians mode, that I just stuck to radians mode, and whenever I need to work with degrees, I just wrote $x\times\frac{\pi}{180}$ instead of $x^\circ$. – Joe Jun 25 '23 at 13:06
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I'm almost always using radians, but this calculator resets back to degrees whenever I reset the memory to clear the variables. – no1dea Jun 25 '23 at 13:13