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GregA
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Homework Statement
What I'm scratching my head with is trying to find mechanically, a way of finding all the solutions of the expression [tex]sin(a \theta ) - sin(b \theta) = 0[/tex] (where a and b are integers (for now) with a > b)
Homework Equations
I can get the roots that are regular rotations from a specific starting angle by first solving [tex]a \theta = \pi - b \theta[/tex] and then by letting all subsequent values of [tex] \theta = \frac{(1+2k) \pi}{a+b}[/tex] (where k is any integer)...it is also easy to spot that [tex]\theta = n \pi[/tex] (where n is an integer)
The Attempt at a Solution
what I'm having trouble with is finding quickly the sneaky roots such as [tex]\theta = \frac{2 \pi}{3} or \frac{4 \pi}{3} [/tex] in the expression [tex] sin4 \theta - sin \theta = 0[/tex](amongst others, though this is the question that I started with) that aren't found with the above method.
I have tried to expand [tex] sin4 \theta - sin \theta = 0[/tex] but after:
[tex] 2sin2\theta cos2\theta - sin\theta = 0[/tex]
[tex] 4sin\theta cos\theta (cos^2\theta - sin^2\theta ) - sin\theta = 0[/tex]
[tex] 4cos\theta (2cos^2\theta - 1 ) -1 = 0[/tex]
[tex] 8cos^3\theta - 4cos\theta - 1 = 0[/tex] I can't see a simple way of solving it without using either Newton's method, the cubic formula (there must surely be a simpler way than that though), or getting the protractor out and searching for them.
I'm betting there is a simple method and either I just can't remember it or have not come across it yet...can anyone throw me some hints?
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