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

1 Answer
Aug 30, 2016

(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..