Mathematics

Problem

Without using a calculator, show that the sums of the squares of the sines of

30^\circ,\;45^\circ

and

60^\circ

is equal to the corresponding sum for the cosines. a. (i) Find the exact values of

\sin 30^\circ,\; \sin 45^\circ

and

\sin 60^\circ

, and hence compute

\sin^2 30^\circ,\; \sin^2 45^\circ

and

\sin^2 60^\circ

. b. (ii) Find the exact values of

\cos 30^\circ,\; \cos 45^\circ

and

\cos 60^\circ

, and hence compute

\cos^2 30^\circ,\; \cos^2 45^\circ

and

\cos^2 60^\circ

. c. (iii) Using the results of parts (i) and (ii), show that

\sin^2 30^\circ + \sin^2 45^\circ + \sin^2 60^\circ = \cos^2 30^\circ + \cos^2 45^\circ + \cos^2 60^\circ.