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3*sin(2*a) = -3/sqrt(2)
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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)