Mathematics

Problem

Arrange the following four numbers in ascending order without calculating their decimal values:

\sin 20^{\circ}

\cos 20^{\circ}

\sin 70^{\circ}

and

\cos 70^{\circ}

. Show all necessary reasoning. a. (i) First compare each pair of complementary ratios

\sin 20^{\circ}

and

\cos 70^{\circ}

;

\cos 20^{\circ}

and

\sin 70^{\circ}

, stating which member of each pair is larger. b. (ii) Using the conclusions of part (i) together with the behaviour of

\sin\theta

and

\cos\theta

(0^{\circ},90^{\circ})

, write the complete ascending order of the four given numbers, giving clear justification for every comparison made.