Question

In lecture 3, there's a proof of the derivative of sin(x) at the end. What I don't understand is why theta and the angle in the smaller triangle are the same. Could someone point me to some geometry or math property I'm missing here ?

Answer 1

There is no math property involved over here.
When you rotate arms OP & OQ by 90 degrees, OP becomes parallel PR and OQ becomes parallel to PQ. Hence the angle between OP and OQ (which is theta) is equal to angle between PR and PQ.

Answer 2

I think the "hey, look! turn by 90 and this is that" argument is not very clear (at least not to me).

but we can use other ideas ( I think there are more than one way to show this).