#(x,y) = ( 9 cos(pi/12), 9 sin(pi/12) ) #
#pi/12 = 180^circ/12 = 15^circ #
We can get the trig functions of #15^circ# from the half angle formula or the difference angle formula.
#cos 15^circ = cos( 1/2 (30^circ)) = cos(45^circ - 30^circ)#
The difference angle form avoids nested square roots; let's use that.
#cos 15^circ = cos 45^circ cos 30^circ + sin 45^circ sin 30^circ
#
#cos 15^circ = (sqrt{2}/2)(sqrt{3}/2) + (sqrt{2}/2)(1/2)
#
#cos 15^circ = 1/4 (sqrt{6} + sqrt{2})#
Similarly,
#sin 15^circ = sin 45^circ cos 30^circ - cos 45^circ sin 30^circ #
#sin 15^circ = 1/4( \sqrt{6} - sqrt{2})#
Putting it together, #P(9,{1π}/12)# in rectangular form is:
#( 9 cos(pi/12), 9 sin(pi/12) ) = ( 9/4 (sqrt{6} + sqrt{2}), 9/4 (sqrt{6} - sqrt{2}) ) #
