SOLUTION: 60 find all the solution in the interval 0,2pi sin(x+pi/6)-sin(x-pi/6)=1/2

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Question 174444: 60 find all the solution in the interval 0,2pi
sin(x+pi/6)-sin(x-pi/6)=1/2
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
sin%28x%2Bpi%2F6%29-sin%28x-pi%2F6%29=1%2F2 Start with the given equation


%28sin%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29%29-sin%28x-pi%2F6%29=1%2F2 Expand sin%28x%2Bpi%2F6%29


%28sin%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29%29-%28sin%28x%29cos%28pi%2F6%29-cos%28x%29sin%28pi%2F6%29%29=1%2F2 Expand sin%28x-pi%2F6%29


sin%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29-sin%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29=1%2F2 Distribute


sin%28x%29%28sqrt%283%29%2F2%29%2Bcos%28x%29%281%2F2%29-sin%28x%29%28sqrt%283%29%2F2%29%2Bcos%28x%29%281%2F2%29=1%2F2 Evaluate cos%28pi%2F6%29 to get sqrt%283%29%2F2. Evaluate sin%28pi%2F6%29 to get 1%2F2


%28sqrt%283%29%2F2%29sin%28x%29%2B%281%2F2%29cos%28x%29-%28sqrt%283%29%2F2%29sin%28x%29%2B%281%2F2%29cos%28x%29=1%2F2 Rearrange the terms.


cos%28x%29=1%2F2 Combine like terms.


cos%28x%29=arccos%281%2F2%29 Take the arccosine of both sides


x=pi%2F3 or x=-pi%2F3 Take the arccosine of 1/2 to get pi%2F3 or -pi%2F3


Since the value x=-pi%2F3 is NOT in the range [0,2pi), this means that -pi%2F3 is NOT a solution


So the first solution is x=pi%2F3



Now add 2pi to pi%2F3 to get 2pi%2Bpi%2F3=6pi%2F3%2Bpi%2F3=%287pi%29%2F3. Take note how this value is outside the interval [0,2pi). So this value is NOT a solution


Now add 2pi to -pi%2F3 to get 2pi-pi%2F3=6pi%2F3-pi%2F3=%285pi%29%2F3. Since this value is inside the interval [0,2pi), this means that %285pi%29%2F3 is a solution



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Answer:


So the solutions are x=pi%2F3 or %285pi%29%2F3