Use double angle identities to evaluate trig functions

Question

1. if sin a = 5/13 and 0 < a < pi/2, find sin 2a , cos 2a, tan 2a
2. if tan a = 1/2 and 0< a < pi/2 , find sin2a , cos2a, tan2a

Answer 1

sin a = 5/13 ---> cos a = 12/13 because the right triangle is opposite=5, hypotnuese=13, which forces

adjacent=12

tan a = sin a / cosa = 5/13 divided by 12/13 = 5/12

sin 2a = 2 sin a cos a = 2(5/13)(12/13) = 120/169

cos 2a = cos^2 - sin^2 = (12/13)^2 - (5/13)^2 = 144/169 - 25/169 = 119/169

tan 2a = 2 tan A / (1 - tan^2) = 2 (5/12) divided by 1 - (5/12)^2 =

5/6 divided by 1 - 25/144 =

5/6 divided by (144-25)/144 =

5/6 * 144/ 119 =

720 / 714 =

360/357 =

120/119