SOLUTION: Please help me on the following question: Find the exact value of sin(-10π/3)

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Question 190623: Please help me on the following question:
Find the exact value of sin(-10π/3)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Here's the unit circle for reference:





sin%28-10pi%2F3%29 Start with the given expression.


-sin%2810pi%2F3%29 Use the identity sin%28-x%29=-sin%28x%29


Take note that 10pi%2F3=2pi%2B4pi%2F3


-sin%282pi%2B4pi%2F3%29 Break up 10pi%2F3 to get 2pi%2B4pi%2F3


-%28sin%282pi%29cos%284pi%2F3%29%2Bcos%282pi%29sin%284pi%2F3%29%29 Use the identity sin%28x%2By%29=sin%28x%29cos%28y%29%2Bcos%28x%29sin%28y%29 to expand


-%28%280%29cos%284pi%2F3%29%2B%281%29sin%284pi%2F3%29%29 Take the sine of 2pi to get 0. Take the cosine of 2pi to get 1 (use the unit circle).


-%280%2Bsin%284pi%2F3%29%29 Multiply


-sin%284pi%2F3%29 Simplify



Take note that 4pi%2F3=pi%2Bpi%2F3



-sin%28pi%2Bpi%2F3%29 Break up 4pi%2F3 to get pi%2Bpi%2F3


-%28sin%28pi%29cos%28pi%2F3%29%2Bcos%28pi%29sin%28pi%2F3%29%29 Use the identity sin%28x%2By%29=sin%28x%29cos%28y%29%2Bcos%28x%29sin%28y%29 to expand


-%28%280%29cos%28pi%2F3%29%2B%28-1%29sin%28pi%2F3%29%29 Take the sine of pi to get 0. Take the cosine of pi to get -1 (use the unit circle).


-%280-sin%28pi%2F3%29%29 Multiply


sin%28pi%2F3%29 Simplify


sqrt%283%29%2F2 Evaluate the sine of pi%2F3 to get sqrt%283%29%2F2 (use the unit circle).



So sin%28-10pi%2F3%29=sqrt%283%29%2F2