Question 871408: please help me solve sin(x) = sin(2x) interval [0,2pi]
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Solve for x
sin(x) = sin(2x)
sin(x) = 2*sin(x)*cos(x)
0 = 2*sin(x)*cos(x) - sin(x)
2*sin(x)*cos(x) - sin(x) = 0
sin(x)(2*cos(x) - 1) = 0
sin(x) = 0 or 2*cos(x) - 1 = 0
sin(x) = 0 or 2*cos(x) = 1
sin(x) = 0 or cos(x) = 1/2
x = arcsin(0) or x = arccos(1/2)
x = pi*n or x = pi/3 + 2pi*n or x = -pi/3 + 2pi*n
All of the solutions can be generated by any one of these three equations (they combine to form a collective set of solutions). The n is an integer. So if you wanted one solution, you could plug in n = 1 into any of the three equations below
x = pi*n or x = pi/3 + 2pi*n or x = -pi/3 + 2pi*n
Let's plug in whole numbers into each equation to generate the solutions. Keep in mind we restrict the interval to [0,2pi]
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For the first general solution x = pi*n:
If n = 0, then
x = pi*n
x = pi*0
x = 0 ... in the interval [0,2pi] = [0,6.28], so we keep it
If n = 1, then
x = pi*n
x = pi*1
x = pi
x = 3.14 ... in the interval [0,2pi] = [0,6.28], so we keep it
If n = 2, then
x = pi*n
x = pi*2
x = 2pi
x = 6.28 ... in the interval [0,2pi] = [0,6.28], so we keep it
We stop here since we've reached the boundary
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For the second general solution x = pi/3 + 2pi*n:
If n = 0, then
x = pi/3 + 2pi*n
x = pi/3 + 2pi*0
x = pi/3
x = 1.0471975511966 ... in the interval [0,2pi] = [0,6.28], so we keep it
If n = 1, then
x = pi/3 + 2pi*n
x = pi/3 + 2pi*1
x = 7pi/3
x = 7.33038285837619 ... not in the interval [0,2pi] = [0, 6.28], so we toss this solution
We stop here
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For the third general solution x = -pi/3 + 2pi*n:
If n = 0, then
x = pi/3 + 2pi*n
x = -pi/3 + 2pi*0
x = -pi/3
x = -1.0471975 ... not in the interval [0,2pi], so we toss this solution
If n = 1, then
x = pi/3 + 2pi*n
x = -pi/3 + 2pi*1
x = 5pi/3
x = 5.23598775598299 ... in the interval [0,2pi], so we keep it
If n = 2, then
x = pi/3 + 2pi*n
x = -pi/3 + 2pi*2
x = 11pi/3
x = 11.5191730631626 ... not in the interval [0,2pi], we toss/ignore it
We stop here
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Final Answer: Below are the solutions to sin(x) = sin(2x) in the interval [0, 2pi]
0, pi, 2pi, pi/3, 5pi/3
The sorted solutions (smallest to largest) are:
0, pi/3, pi, 5pi/3, 2pi
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