1. Trig Identities An identity is a statement which is true for all values. eg 3x(x + 4) = 3x 2 + 12x eg(a + b)(a – b) = a 2 – b 2 Trig Identities (1)sin 2  + cos 2  = 1 (2)sin  = tan  cos    an odd multiple of  /2 or 90°.

2. Reason  p q r p 2 = q 2 + r 2 sin  = r / p cos  = q / p tan  = r / q ************** (1) sin 2  + cos 2  = ( r / p ) 2 + ( q / p ) 2 = r 2 + q 2 p 2 p 2 = q 2 + r 2 p 2 = p 2 p 2 = 1

3. (2) sin  cos  = r / p  q / p = r / p X p / q = pr / pq = r / q = tan  ****************** NB:since sin 2  + cos 2  = 1 then sin 2  = 1 - cos 2  and cos 2  = 1 - sin 2 

4. Example1 sin  = 5 / 13 where 0 <  <  / 2 Find the exact values of cos  and tan . ******************** NB:NO CALCS!!!! cos 2  = 1 - sin 2  = 1 – ( 5 / 13 ) 2 = 1 – 25 / 169 = 144 / 169 cos  =  ( 144 / 169 ) = 12 / 13 or - 12 / 13 Since  in Q1 then cos  > 0 Socos  = 12 / 13 tan  = sin  cos  = 5 / 13  12 / 13 = 5 / 13 X 13 / 12 ie tan  = 5 / 12

5. Example 2 Given that cos  = - 2 /  5 where  <  < 3  / 2 find sin  and tan . **************** sin 2  = 1 - cos 2  = 1 – (- 2 /  5 ) 2 = 1 – 4 / 5 = 1 / 5 sin  =  ( 1 / 5 ) = 1 /  5 or - 1 /  5  In Q3 so sin  < 0 Hence sin  = - 1 /  5 tan  = sin  cos  = - 1 /  5  - 2 /  5 = - 1 /  5 X -  5 / 2 Hence tan  = 1 / 2