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December 10th, 2012, 09:59 PM   #1
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Complex numbers: Polar to Cartesian Form

I really need help here!

Using the triple angle identities for sin, cos and tan, Solve the equation
1.) 4x^3 - 3x = -(1/sqrt2)
At first instinct I saw that it resembled the triple angle identity for cosine so I had cos3x=-(1/sqrt2), arccos(1/sqrt2) gave me pi/4 and then cosx is pi/12. However, the answer had cos(11pi/12) and cos(5pi/12), how did they get that?

2.) Solve x^3 - 3sqrt(3)x^2 -3x + sqrt(3) = 0. I couldn't see how it represented any of the triple identities. I thought tan3x is possible but I have problems with the coefficients, they don't tally!

Please help me, thank you so very much!
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December 11th, 2012, 06:03 AM   #2
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(1). Let x = cos(u), then cos(3u) = -1/?2.
Hence 3u = 3?/4 + 2k?, where k is an integer.
Three possibilities are 3u = 3?/4 or -3?/4+2? or 3?/4+2?, giving u = ?/4 or 5?/12 or 11?/12.
Hence x = cos(u) = 1/?2 or 1/(?2 + ?6) or 1/(?2 - ?6).

(2). The equation can be rearranged as (3x - x)/(1 - 3x) = ?3. Let x = tan(u), then tan(3u) = ?3, etc.
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