Question

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

Answer 1

`(3+4sqrt 3)/10`.

Explanation:

Let `a = sin^(-1)(1/2) in Q1, for the principal value`. Then,

`sin a = 1/2 and cos a = sqrt(1-sin^2 a) = sqrt(1-1/4) =sqrt 3/2`, for a in

Q1.

Let `b = cos^(-1)(3/5) in Q1, for the principal value`. Then,

`cosb = 3/5 and sin b = sqrt(1-cos^2 a) = sqrt(1-9/25) =4/5`, for b in

Q1.

Now, the given expression is

`sin(a+b)`

`=sina cos b +cos a sin b`

`=(1/2)(3/5)+(sqrt 3/2)(4/5)`

`(3+4sqrt 3)/10`.

Yet, a could be in Q3, wherein `cos a = - sqrt 3/2`, and, similarly, b

could be in Q4, wherein `sin b = -4/5`. Considering this, the

general value is

`+-3+-4sqrt 3)/10`,

when the principal-value convention is relaxed..