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If I change units of an angle for radians to degrees in the next expresion $$4\sin (\theta) \frac{d \theta}{d t}=\frac{dy}{dt}$$ the value of $$\frac{dy}{dt}$$ changes.

For example at a rate of change of $\frac{d\theta}{dt} = 30deg , \qquad $ and $\frac{d\theta}{dt}=\frac{\pi }{6}rad$ the rate of change is the same, but the final expresion is not.

So which is the correct unit? and mathematicaly why is the reason?.

I already know that the correct unit are radians, Im looking for a more formal and deeper explanation of why this units are the correct.

Thanks

JuanMuñoz
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1 Answers1

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For dimensional consistency, firstly, I would expect the number '4' to be dimensionfull. Additionally, whether the angle is to be taken in radians or in degrees depends on where this equation came from. If I venture a guess, then I suppose at some point you differentiated a relation between $y$ and $\theta$. In that case, the relation you started of with will specify the units for $\theta$ after which the differentiation would have to carried out accordingly. Remember, $$ \dfrac{\mathrm{d}}{\mathrm{d}\theta}\cos{\theta}=-\sin\theta\\ \dfrac{\mathrm{d}}{\mathrm{d}\theta}\cos{\theta^{\circ}}=-\dfrac{\pi}{180}\sin\theta^{\circ}\\ \dfrac{\mathrm{d}}{\mathrm{d}\theta^{\circ}}\cos{\theta^{\circ}}=-\sin\theta^{\circ}\\ $$

surajshankar
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