1. The problem statement, all variables and given/known data
Use de Moivre's Theorem to derive an expression in terms of sines and cosines for sin 3x and cos 3x.
Hence deduce that ##\tan 3x=\frac{3\tan x-\tan^3 x}{1-3\tan^2 x}##
Use the result above to solve the equation ##t^{3}-3t^{2} -3t+1=0
2. Relevant equations
de Moivre's theorem
3. The attempt at a solution
I have compared the real and imaginary parts of ##(cos 3x+ i sin x)## with ##(cos x+i sin x)^{3}##
Then ##cos 3x=cos^3 x-3 cos x sin^2 x## and ##sin 3x=3cos^2 x sin x-sin^3 x##
I have proved that tan 3x identity, but how do I solve the equation?
I know I have to use set tan 3x=1, but what is the domain of x?
Use de Moivre's Theorem to derive an expression in terms of sines and cosines for sin 3x and cos 3x.
Hence deduce that ##\tan 3x=\frac{3\tan x-\tan^3 x}{1-3\tan^2 x}##
Use the result above to solve the equation ##t^{3}-3t^{2} -3t+1=0
2. Relevant equations
de Moivre's theorem
3. The attempt at a solution
I have compared the real and imaginary parts of ##(cos 3x+ i sin x)## with ##(cos x+i sin x)^{3}##
Then ##cos 3x=cos^3 x-3 cos x sin^2 x## and ##sin 3x=3cos^2 x sin x-sin^3 x##
I have proved that tan 3x identity, but how do I solve the equation?
I know I have to use set tan 3x=1, but what is the domain of x?
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