#(sin x)'=cos x, (cos x)'=-sin x, int sin x dx = -cos x + C, int cos x dx = sin x + C and e^(ix)=cos x + i sin x#. Are all these OK, if x is in degree mode?

Question

#(sin x)'=cos x, (cos x)'=-sin x, int sin x dx = -cos x + C, int cos x dx = sin x + C and e^(ix)=cos x + i sin x#. Are all these OK, if  x is in degree mode?

Serenity Jones · Accepted Answer

correct.

Solution is correct.

Anna Allen · Answer

undefined No, those statements are not entirely accurate if $ x $ is in degree mode. The derivatives and integrals you provided are correct for trigonometric functions when $ x $ is in radian mode. However, if $ x $ is in degree mode, the derivatives and integrals need to be adjusted accordingly. Here are the correct versions:

1. $ \frac{d}{dx}(\sin x) = \cos x $ holds true regardless of whether $ x $ is in degree or radian mode.
2. $ \frac{d}{dx}(\cos x) = -\sin x $ holds true regardless of whether $ x $ is in degree or radian mode.
3. $ \int \sin x \, dx = -\cos x + C $ holds true regardless of whether $ x $ is in degree or radian mode.
4. $ \int \cos x \, dx = \sin x + C $ holds true regardless of whether $ x $ is in degree or radian mode.
5. $ e^{ix} = \cos x + i \sin x $ holds true only when $ x $ is in radian mode. If $ x $ is in degree mode, you need to convert degrees to radians before using this formula.