Question

Evaluate: `cos (sin^-1 (4/5) + sin^-1 (12/13))`

Answer 1

`sqrt(5^2 - 4^2)` = 3

cos A = `"Adj"/"Hyp" = 3/5`

Let `sin^-1 (4/5)` = A

sin A = `4/5`

∴ cos A = `3/5`

`sqrt(169 - 144) = sqrt 25` = 5

cos B = `"Adj"/"Hyp" = 5/13`

Let `sin^-1 (12/13)` = B

`12/13` = sin B

sin B = `12/13`

∴ cos B = `5/13`

Now `cos (sin^-1 (4/5) + sin^-1 (12/13))` = cos (A + B)

= cos A cos B – sin A sin B

`= 3/5 xx 5/13 - 4/5 xx 12/13`

`= 15/65 - 48/65`

`= - 33/65`