Given: sin a=3/5 and sin b= 5/13, 0 < a < pi/2 and 0 < b < pi/2 Find cos( a+b )

Question

answers are

63/65

33/65

14/65

Answer 1

if sina =3/5, cosa = 4/5, if sinb = 5/13, cosb = 12/13

use the sum and difference formulas

cos(a+b) = cosacosb - sinasinb = (4/5)(12/13) - (3/5)(5/13) = 48/65 - 15/65 = 33/65

to check the answer use a calculator with an inverse trig function

if sina =3/5 then a = about 36.8 degrees

if sinb = 5/13 then b = about 22.62 degrees

cos(36.8+22.62) = cos(59.42) = about .509

33/65 = about .508

could be some rounding approximations, to err on the 3rd decimal place,= but cos(a+b) == exactly 33/65 = about .508