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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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