# Units in related rates problems

What is the unit of $\frac{d\theta}{dt}$?

Let's say I have a triangle with an angle $\theta$ and the opposite and hypotenuse sides $y$ and $h$, respectively, so that

$\sin \theta$ = $\frac{y}{h}$

Taking the derivative of both sides with respect to $t$:

$\cos \theta$ $\frac{d\theta}{dt}$ = $\dfrac{\frac{dy}{dt}h-\frac{dh}{dt}y}{h^2}$

The units of the right side, where $m$ is distance (meters) and $s$ is time (seconds):

$\dfrac{\dfrac{m}{s}m-\dfrac{m}{s}m}{m^2}$ = $\dfrac{1}{s}$

Since $\cos \theta$ is a ratio, it has no units, therefore:

$\dfrac{d\theta}{dt}$ = $\dfrac{1}{s}$

However, by common sense, $\dfrac{d\theta}{dt}$ should be in $\dfrac{^o}{s}$ (if using degrees).

Why is it different, and if I am making a mistake, how could it be corrected?

• Edited to take out the part about it not making a difference in this case. It will make a numerical difference. If your rate is given as (say) a steady increase in angle of $30$ degrees per second, and you're asked for the rate of change of $\sin \theta$ at the instant when the angle is $60$ degrees, then the correct answer is not simply $\frac{1}{2}\cdot (30) = 15$ (and what units?), but rather $\frac{1}{2} \cdot \frac{\pi}{6} = \frac{\pi}{12}$, and the unit is $s^{-1}$. – Deepak Oct 3 '14 at 1:11