Trig Functions

STEP 1998, Math II, #7: Let

(i) By considering and , show that for .

(ii) Show similarly that for .

(iii) Show that for , and hence that

for .

(iv) By considering , show that for .

STEP 2002, Math I, #3: Show that .

and are constants. State, with brief reasons, which points are maxima and which are minima. Hence prove that

if and only if for some integer .

Show also that for all values of and deduce that there are no solutions to the equation .

Sketch, on the same axes, the graphs of and . Sketch, not on the previous axes, the graph of .