Question 924330
Part a)



Solve for {{{theta}}}



{{{2*cos(2theta)-1 = 0}}}



{{{2*cos(2theta)= 1}}}



{{{cos(2theta)= 1/2}}}



{{{2theta= arccos(1/2)}}}



{{{2theta= pi/3+2pi*k}}} or {{{2theta= -pi/3+2pi*k}}} where k is any integer.



{{{theta= (1/2)*(pi/3+2pi*k)}}} or {{{theta= (1/2)*(-pi/3+2pi*k)}}}



{{{theta= pi/6+pi*k}}} or {{{theta= -pi/6+pi*k}}}



The general solutions are {{{theta= pi/6+pi*k}}} or {{{theta= -pi/6+pi*k}}} where k is any integer.



You can condense that into one equation: {{{theta= ""+-pi/6+pi*k}}} (take note of the plus/minus)



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Part b)



We'll find the first set of solutions.



Plug in integer values of k into {{{theta= pi/6+pi*k}}} to generate solutions. Make sure you are between {{{x = 0}}} and {{{x = 2pi = 6.28}}} roughly.



Plug in {{{k =  0}}}



{{{theta= pi/6+pi*k}}}



{{{theta= pi/6+pi*0}}}



{{{theta= pi/6}}}



{{{theta= 0.5235987755983}}}



This is in the interval [0,2pi), so we keep it.



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Plug in {{{k =  1}}}



{{{theta= pi/6+pi*k}}}



{{{theta= pi/6+pi*1}}}



{{{theta= pi/6+pi}}}



{{{theta= 7pi/6}}}



{{{theta= 3.6651914291881}}}



This is in the interval [0,2pi), so we keep it.



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Plug in {{{k =  2}}}



{{{theta= pi/6+pi*k}}}



{{{theta= pi/6+pi*2}}}



{{{theta= 6.8067840827779}}}



we have gone over 6.28, so we stop here and we don't include this solution because it is not in the interval [0,2pi)



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Now we move onto the second part.



Plug in integer values of k into {{{theta= -pi/6+pi*k}}} to generate solutions. Make sure you are between {{{x = 0}}} and {{{x = 2pi = 6.28}}} roughly.



Plug in {{{k =  0}}}



{{{theta= -pi/6+pi*k}}}



{{{theta= -pi/6+pi*0}}}



{{{theta= -0.5235987755983}}}



That is less than x = 0, so we toss out this solution.



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Plug in {{{k =  1}}}



{{{theta= -pi/6+pi*k}}}



{{{theta= -pi/6+pi*1}}}



{{{theta= -pi/6+pi}}}



{{{theta= 5pi/6}}}



{{{theta= 2.6179938779915}}}



This is in the interval [0,2pi), so we keep it.



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Plug in {{{k =  2}}}



{{{theta= -pi/6+pi*k}}}



{{{theta= -pi/6+pi*2}}}



{{{theta= -pi/6+2pi}}}



{{{theta= 11pi/6}}}



{{{theta= 5.7595865315813}}}



This is in the interval [0,2pi), so we keep it.



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Plug in {{{k =  3}}}



{{{theta= -pi/6+pi*k}}}



{{{theta= -pi/6+pi*3}}}



{{{theta= 8.9011791851711}}}



we have gone over 6.28, so we stop here and we don't include this solution because it is not in the interval [0,2pi)



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The exact solutions to part b) are {{{x = pi/6}}}, {{{x = 7pi/6}}}, {{{x = 5pi/6}}}, {{{x = 11pi/6}}}. They are exact solutions in terms of {{{pi}}}



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Let me know if that helps or not. Thanks.


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