Question

How do you simplify `sin ( sin^ -1 (-3/5) + tan^ -1(5/12))`?

Answer 1

`sin(sin^(-1)(-3/5)+tan^(-1)(5/12))=-16/65`

Explanation:

Let `sin^(-1)(-3/5)=alpha` and `tan^(-1)(5/12)=beta`

then `sinalpha=-3/5` and `tanbeta=5/12`

and `cosalpha=sqrt(1-(-3/5)^2)=sqrt(1-9/25)=sqrt(16/25)=4/5`

`cosbeta=1/(sqrt(1+tan^2beta))=1/(sqrt(1+(5/12)^2))=1/(sqrt(1+25/144))=1/(sqrt(169/144))=1/(13/12)=12/13`

and `sinbeta=tanbetaxxcosbeta=5/12xx1/13=5/13`

Hene, `sin(sin^(-1)(-3/5)+tan^(-1)(5/12))=sin(alpha+beta)`

= `sinalphacosbeta+cosalphasinbeta`

= `(-3/5)xx12/13+4/5xx5/13`

= `-36/65+20/65=-16/65`