Question

Evaluate cos(sin^-1(4/5)

Answer 1

Explanation:

`Let \: \: \theta = \sin^( - 1) ((4)/(5)) \\ \\ \therefore \: \sin\theta = (4)/(5) \\ \\ \because \: { \cos}^(2) \theta =1 - { \sin}^(2) \theta \\ \\ \therefore \: { \cos}^(2) \theta = 1 - ((4)/(5) )^(2) \\ \\ = 1 - (16)/(25) \\ \\ = (25 - 16)/(25) \\ \\ = (9)/(25) \\ \\ \therefore \:{ \cos}\theta = \pm \: (3)/(5) \\ \\ \because \: \theta \: lie \: in \: the \: first \: quadrant \\ \\ \therefore \: { \cos}\theta = \: (3)/(5) \\ \\ \implies \theta = { \cos}^( -1 ) \: (3)/(5)\\\\\implies \theta = { \cos}^( -1 ) \: (3)/(5)= { \sin}^( -1 ) \: (4)/(5) \\ \\ \therefore \: \cos( \ {sin}^( - 1) (4)/(5) ) \\ \\ = \cos( \ {cos}^( - 1) (3)/(5) ) \\\\ = (3)/(5) \\ \\ \purple{ \boxed{\therefore \: \cos( \ {sin}^( - 1) (4)/(5) ) = (3)/(5)}}`