Let a = sin^(-1)(1/2) in Q1, for the principal valuea=sin−1(12)∈Q1,forthepr∈cipalvalue. Then,
sin a = 1/2 and cos a = sqrt(1-sin^2 a) = sqrt(1-1/4) =sqrt 3/2sina=12andcosa=√1−sin2a=√1−14=√32, for a in
Q1.
Let b = cos^(-1)(3/5) in Q1, for the principal valueb=cos−1(35)∈Q1,forthepr∈cipalvalue. Then,
cosb = 3/5 and sin b = sqrt(1-cos^2 a) = sqrt(1-9/25) =4/5cosb=35andsinb=√1−cos2a=√1−925=45, for b in
Q1.
Now, the given expression is
sin(a+b)sin(a+b)
=sina cos b +cos a sin b=sinacosb+cosasinb
=(1/2)(3/5)+(sqrt 3/2)(4/5)=(12)(35)+(√32)(45)
(3+4sqrt 3)/103+4√310.
Yet, a could be in Q3, wherein cos a = - sqrt 3/2cosa=−√32, and, similarly, b
could be in Q4, wherein sin b = -4/5sinb=−45. Considering this, the
general value is
+-3+-4sqrt 3)/10±3±4√3)10,
when the principal-value convention is relaxed..
