Question

How do you simplify `sin(sin^-1(1/2)+cos^-1(3/5))`?

Answer 1

`sin(sin^(-1)(1/2)+cos^(-1)(3/5))=color(green)(1.3)`

Explanation:

`sin^(-1)` is by definition an angle in `[0,pi)`
within this interval `sin(sin^(-1)(theta))=theta`
`color(white)("XXX")rarr sin(sin^(-1)(1/2))=1/2`

`cos^(-1)` is by definition an angle in `(-pi/2,pi/2]`
within this range a `cos=y/r` of `3/5` implies a standard triangle with
`color(white)("XXX")x/r=sin=4/5`
So `cos^(-1)(3/5)=sin^(-1)(4/5)`
and, again, since this is in QI,
`color(white)("XXX")sin(cos^(-1)(3/5))=sin(sin^(-1)(4/5))=4/5`

Therefore
`sin(sin^(-1)(1/2)+cos^(-1)(3/5))`
`color(white)("XXX")=1/2+4/5`

`color(white)("XXX")=13/10 (=1 3/10 = 1.3)`