Consider on of those rising balloon related rates Calc problems. Based on the actual problem, you'd label a triangle with a few sides and one of the angles as $\theta$. You set up some trig relationship, and then take the derivative. $$tan(\theta)=\frac{y}{100}$$
$$ sec^2(\theta)\frac{d\theta}{dt}=\frac{1}{100}\frac{dy}{dt}$$
Right there, now that we have a $\frac{d\theta}{dt}$, there is an implied unit, since this refers to the rate of change of the angle SIZE. Is the unit degrees or radians? We never actually refer to a specific size, so it stays ambiguous.
Sub in some givens, and get something like $$\frac{d\theta}{dt}=7 \frac{rad}{min}$$
Yet, nowhere in the problem have I actually referred to the actual size of $\theta$. All we ever dealt with was the trig relationship of the sides. So, can't it be measured in either radians or degrees?
So, my question is why can't the solution just as be $$\frac{d\theta}{dt}=7 \frac{degrees}{min}??$$ Who ever stated it's measure was in radians? Is this other link related? Something to do with implied radians when you take the derivative?
Best Answer
You are correct that when you write $\tan \theta = \frac y{100}$ there is nothing with a scale and you can measure the angle in degrees or radians. When you write $\frac {d\tan \theta}{d\theta}=\sec^2 \theta$ you are measuring the change in angle in some unit. Radians are the natural one and the one we prove this identity (as well as all the others) in. You could write $\frac {d\tan \theta^\circ}{d\theta^\circ}=\frac {\pi}{180}\sec^2 \theta^\circ$ if you want, where the $\frac \pi{180}$ comes from applying the chain rule.