SOLUTION: find the exact value of this expression: sin(tan^-1(-1)+sin^-1(1/2))

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Question 979784: find the exact value of this expression: sin(tan^-1(-1)+sin^-1(1/2))
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!

Hello,

I read your writing   sin(tan^-1(-1)+sin^-1(1/2))   as

sin%28arctan%28-1%29+%2B+arcsin%281%2F2%29%29.

Do I read correctly?

If my reading is correct,  then the rest part is simple.

arcsin%281%2F2%29 = pi%2F6 = 30°,

arctan%28-1%29 = -pi%2F4 = -45°.

sin(30° - 45°) = sin(30°)*cos(45°) - cos(30°)*sin(45°)       (subtraction formula for sines of  Trigonometry,  see the lesson  Addition and subtraction formulas  in this site)

                      = 1%2F2.sqrt%282%29%2F2 - sqrt%283%29%2F2.sqrt%282%29%2F2 = sqrt%282%29%2F4 - sqrt%286%29%2F4 = %28sqrt%282%29-sqrt%286%29%29%2F4 = -%28sqrt%286%29-sqrt%282%29%29%2F4 = -0.259.