Question 1101623: Determine the values of
2θ (not θ) on [0,2π) that satisfy the following equation. (Separate multiple solutions with a comma. Give exact answers.)
3sin(2θ)=−3/√2
2θ=
You now have two equations representing all possible solutions for 2θ. Solve each of those equations for θ. (Let θ1 and θ2 represent the solutions on [0,2π), where θ1is less than θ2.)
θ1=
θ2=
Use these general solutions for θ to find the four solutions to 3sin(2θ)=−3/√2 on the intervall
[0,2π). (Separate multiple solutions with a comma. Give exact answers.)
θ=
Found 2 solutions by stanbon, ikleyn:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Determine the values of 2t (not t) on [0,2π) that satisfy the following equation. (Separate multiple solutions with a comma. Give exact answers.)
3sin(2t)=−3/√2
sin(2t) = -1/sqrt(2)
2t = 5pi/4 or 2t = 7pi/4
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You now have two equations representing all possible solutions for 2θ. Solve each of those equations for θ. (Let θ1 and θ2 represent the solutions on [0,2π), where θ1is less than θ2.)
t1= 5pi/8
t2= 7pi/8
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Cheers,
Stan H.
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Use these general solutions for θ to find the four solutions to 3sin(2θ)=−3/√2 on the interval
[0,2π). (Separate multiple solutions with a comma. Give exact answers.)
θ=
Answer by ikleyn(52818)  (Show Source):
You can put this solution on YOUR website! .
3*sin(2*a) = -3/sqrt(2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Due to "typography" issues, I will replace  in my post by simple "a".
 =  (1) ====> (divide both sides by 3) ====>
 =  , or, which is the same,
=  .
It implies  =  or  =  .
Everything was simple to this point.
But in reality, accurate analysis only STARTS from this point.
1) It is obvious that = implies  =  .
But if you stop here, you will loose another existing solution of the same family.
It is  =  =  .
Indeed,  =  =  is GEOMETRICALLY the same angle as and has the same value of sine,
so is the solution to the original equation (1), too.
Thus the relation = creates and generates not one solution , but TWO solutions and
of the same family. Notice, that they BOTH belong to the interval [0, ).
2) The same or the similar story is with the solution = .
It is obvious that = implies =  .
But if you stop here, you will loose another existing solution of the same family.
It is  =  =  .
Indeed,  =  =  is GEOMETRICALLY the same angle as and has the same value of sine,
so is the solution to the original equation (1), too.
Thus the relation = creates and generates not one solution , but TWO solutions and
of the same family. Notice, that they BOTH belong to the interval [0,).
3. Thus the original equation (1) has 4 (four, FOUR) solutions in the interval [0,):
, , and .
4. The plot below visually confirms existing of 4 solutions to the given equality:
Plot y =  (red) and y = (green)
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