My Math Forum Complex numbers: Polar to Cartesian Form

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 December 10th, 2012, 09:59 PM #1 Senior Member   Joined: Apr 2012 Posts: 112 Thanks: 0 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!
 December 11th, 2012, 06:03 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,927 Thanks: 2205 (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 - x³)/(1 - 3x²) = ?3. Let x = tan(u), then tan(3u) = ?3, etc.

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