Your question about why not the -axis is pointless. A definition is a definition. You may define a different reference angle starting from the -axis if you wish - or indeed any other line - but in doing so, you will be creating a different definition. You might as well ask "Why doesn't the alphabet start at the letter 'n'?". Without some agreed conventions, we should get nowhere.
OK then. If the point is on the unit circle, and the radius joining to makes an angle with the positive direction of the -axis, measured anticlockwise, as described above, then, by definition:This definition holds good for all values of ; i.e. for angles in all quadrants. In particular, in QIII, where and , this will mean that and are both negative.
Draw a diagram, showing an angle between and (i.e. lies in QIII). Now draw the diameter through to meet the circle again in QI. You should be able to see that this shows that:where ; i.e. is an acute angle. This shows how to relate the values of sine and cosine in QIII to those in QI.