A is always less than (or, in particular, equal to ). Let us consider some examples. Example : Given tan A = , find the other trigonometric ratios of the angle A. Solution : Let us first draw a right ABC (see Fig .
). Now, we know that tan A = BC . Therefore, if BC = k , then AB = k , where k is a positive number. Now, by using the Pythagoras Theorem, we have AC = AB + BC = ( k ) + ( k ) = k So, AC = k Now, we can write all the trigonometric ratios using their definitions.
sin A = BC cos A = AB Therefore, cot A = , cosec A = tanA and sec A = Example : If B and Q are acute angles such that sin B = sin Q, then prove that B = Q. Solution : Let us consider two right triangles ABC and PQR where sin B = sin Q (see Fig. . ).
We have sin B = AC and sin Q = PR Fig. . Fig. .
Then AB = PR Therefore, PR = , say ( ) Now, using Pythagoras theorem, BC = and QR = PQ – PR So, QR = PR PR PR PR PR ( ) From ( ) and ( ), we have PR = AB QR Then, by using Theorem . , ACB ~ PRQ and therefore, B = Q. Example : Consider ACB, right-angled at C, in which AB = units, BC = units and ABC = (see Fig. .
). Determine the values of (i) cos + sin , (ii) cos – sin Solution : In ACB, we have AC = ( ) ( ) ( )( ) ( )( ) 20units So, sin = AC , cos Now, (i) cos