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)   (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...!



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.






or
or or

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

theta = 0: ; YES

theta = (4/3)pi: ; YES

theta = (5/3)pi: ; 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:
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.

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


     -  = 1.


Multiply both sides by .

     -  = .


It can be written in this form

     -  = .


Use the formula of adding arguments for sine

     = .


It may happen in one of the two cases


Case 1.   = .

          which implies  = 0,

or

Case 2.   =  - ,

         which implies   =  =  = -.

         Since we consider the angles in the interval [0,2pi), it is equivalent to   = .


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

                         a)   = 0,   and  b)   = .

Solved.

---------------

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


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