The point with coordinates \((a,b)\), where \(a\) and \(b\) are rational numbers, is called:
an integer rational point if both \(a\) and \(b\) are integers;
a non-integer rational point if neither \(a\) nor \(b\) is an integer.
- Write down an integer rational point and a non-integer rational point on the circle \(x^2+y^2=1\).
What are the simplest possible points here? Where does the circle cut the axes?
Could we see a point on the unit circle as a vertex of a right-angled triangle, with the circle’s radius as the hypotenuse?
What right-angled triangles do we know with integer-length sides? Can we use these?
- Write down an integer rational point on the circle \(x^2+y^2=2\). Simplify \[(\cos\theta+\sqrt{m}\sin\theta)^2+(\sin\theta-\sqrt{m}\cos\theta)^2\] and hence obtain a non-integer rational point on the circle \(x^2+y^2=2\).
What must the value of \(m\) be?
What values of \(\cos\theta\) and \(\sin\theta\) did we use in part (a) in our Pythagorean triple right-angled triangle?
