# Thread: Circle sector

1. ## Circle sector

Hi , I just started with radian measure an have this

And now i can't figure out how to do point b) ("Using AOC show that R sin theta = ..." part) , I mean I'm not sure if I understand how I should go about this type of question, why is there sin theta ? using triangle area formula gets me nowhere , any advice how i should approach this particular question?

2. ## Re: Circle sector

Since OT = R and CT = r (because of radii of corresponding circles), OC = OT-CT = R-r
Notice that OCE is right-angled triangle with OC = R-r, CE = r, COE = theta, and OEC = pi/2.
Therefore, sin(COE) = sin(theta) = CE/OC = r/(R-r). Simplifying this yields the required expression.

For the clarity of reading,
OC= OT - CT = $\displaystyle R-r$
Since OCE is right-angle triangle with OEC$\displaystyle \angle = \pi/2$,
$\displaystyle \sin(\theta) =$ CE/OC $\displaystyle = r/(R-r)$
$\displaystyle \frac{\sin(\theta)}{1} = \frac{r}{R-r} \implies \frac{\sin(\theta)}{1-\sin(\theta)} = \frac{r}{R} \implies R\sin(\theta) = r(1+\sin(\theta))$