Testwiki:Reference desk/Archives/Mathematics/2017 July 15

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July 15

Trig function derivative (degrees)

I think I asked this question a few years ago:

What is the derivative of $\arctan \left(\frac{\sin (x)}{\cos (x)+c}\right)$ while $x$ is in degrees? יהודה שמחה ולדמן (talk) 20:51, 15 July 2017 (UTC)

It is the derivative of the function with the argument expressed in radians multiplied by $\pi /180$. Ruslik_Zero 20:56, 15 July 2017 (UTC)

$\frac{d}{dx}\arctan \left(\frac{\sin (\frac{\pi }{180}x)}{\cos (\frac{\pi }{180}x)+c}\right)=\frac{\pi }{180}\left(\frac{c\cos (\frac{\pi }{180}x)+1}{c^2+2c\cos (\frac{\pi }{180}x)+1}\right)$ ? יהודה שמחה ולדמן (talk) 21:37, 15 July 2017 (UTC)

This is an easy consequence of the chain rule. Let $u=\frac{\pi }{180}x$ and calculate $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}=\frac{dy}{du}\frac{\pi }{180}$.--Jasper Deng (talk) 04:09, 16 July 2017 (UTC)

• But if you want the result to also be in degrees, don't you have to multiply this result by $\frac{180}{\pi }$? Then the result is just the unadjusted formula. Loraof (talk) 15:25, 16 July 2017 (UTC)

The function he has is dimensionless, so the units of the derivative should be inverse degrees, which agrees with the expression given above.--Jasper Deng (talk) 17:41, 16 July 2017 (UTC)

Actually the OP's function is the arctan function ("the angle whose tangent is ..."), whose dimensions are degrees or radians. Loraof (talk) 19:17, 16 July 2017 (UTC)

Right, I got the order of functions reversed. The conversion factor you mentioned should be applied at the beginning; then it drops out as you mentioned.--Jasper Deng (talk) 19:52, 16 July 2017 (UTC)