SOLUTION: \cos\theta-\sqrt{3}\sin\theta=1

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Question 1156563: \cos\theta-\sqrt{3}\sin\theta=1
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Don't use that format to present the problem and make us guess what it is...!

cos%28theta%29-sqrt%283%29sin%28theta%29+=+1

In an equation involving sin(theta) and cos(theta), at some point we are almost certainly going to use sin^2+cos^2=1 to get the equation in either sine or cosine alone.

cos%28theta%29+=+1%2Bsqrt%283%29sin%28theta%29
cos%5E2%28theta%29+=+1%2B2sqrt%283%29sin%28theta%29%2B3sin%5E2%28theta%29
1-sin%5E2%28theta%29+=+1%2B2sqrt%283%29sin%28theta%29%2B3sin%5E2%28theta%29
0+=+4sin%5E2%28theta%29%2B2sqrt%283%29sin%28theta%29
0+=+sin%28theta%29%284sin%28theta%29%2B2sqrt%283%29%29
sin%28theta%29+=+0 or sin%28theta%29+=+-sqrt%283%29%2F2
theta+=+0 or theta+=+%284%2F3%29pi or theta+=+%285%2F3%29pi

At one point in our work we squared both sides of the equation; so we need to check for extraneous solutions.

theta = 0: cos%280%29=+1; 1%2Bsqrt%283%29sin%280%29+=+1 YES

theta = (4/3)pi: cos%28%284%2F3%29pi%29+=+-1%2F2; YES

theta = (5/3)pi: cos%28%285%2F3%29pi%29+=+1%2F2; NO

ANSWER: theta = 0; theta = (-4/3)pi

A graph of cos(theta)-sqrt(3)sin(theta)-1, showing zeros at 0 and (4/3)pi:

graph%28400%2C400%2C-pi%2F2%2C5pi%2F2%2C-4%2C2%2Ccos%28x%29-sqrt%283%29sin%28x%29-1%29

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I will show you  TRULY  elegant  method of solution . . . 


    cos%28theta%29 - sqrt%283%29%2Asin%28theta%29 = 1.


Multiply both sides by 1%2F2.

    %281%2F2%29%2Acos%28theta%29 - %28sqrt%283%29%2F2%29%2Asin%28theta%29 = 1%2F2.


It can be written in this form

    sin%28pi%2F6%29%2Acos%28theta%29 - cos%28pi%2F6%29%2Asin%28theta%29 = sin%28pi%2F3%29.


Use the formula of adding arguments for sine

    sin%28%28pi%2F6%29+-+theta%29 = sin%28pi%2F6%29.


It may happen in one of the two cases


Case 1.  %28pi%2F6%29+-+theta = pi%2F6.

          which implies theta = 0,

or

Case 2.  %28pi%2F6%29+-+theta = pi - pi%2F6,

         which implies  theta = %282pi%29%2F6+-+pi = pi%2F3+-+pi = -%282pi%29%2F3.

         Since we consider the angles in the interval [0,2pi), it is equivalent to  theta = %284pi%29%2F3.


So, we get the ANSWER :  The given equation has two solutions in the interval [0,2pi)

                         a)  theta = 0,   and  b)  theta = %284pi%29%2F3.

Solved.

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@greenestamps,  double check your answer:  it  DOES  NOT  correspond to your plot  (and should be corrected (!) )