Maths with David

    9709. P3. Trigonometry Challenge question

    The unit circle is the circle with radius 1 and centre the origin, O.

    N and P are distinct points on the unit circle. N has coordinates (-1,0), and P has coordinates ( cos \theta , sin \theta ) , where - \pi < \theta < \pi . The line NP intersects the y-axis at Q, which has coordinates (0,q).

    (i) Show that q = tan \frac{1}{2} \theta

    (ii) For q \neq 1 , let f1(q) = \frac{1+q}{1-q} . Show that f1(q) = tan \frac{1}{2} ( \theta + \frac{1}{2} \pi) .

    (iii) Let Q1 be the point with coordinates (0,f1(q)) and P1 be the point of intersection (other than N) of the line NQ1 and the unit circle. Describe geometrically the relationship between P and P1.

    (iv) P2 is the image of P under an anti-clockwise rotation about O through angle \frac{1}{3} \pi . The line NP2 intersects the y-axis at the point Q2 with coordinates (0,f2(q)). Find f2(q) in terms of q, for q \neq \sqrt{3}