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Math Help - Two complex number problems!

  1. #1
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    Two complex number problems!

    Hello!

    (First problem) Express sin5x in terms of a) sinx and b) cosx

    Attempt: According to de Moivre's formula - (cosx+isinx)^n=cos(nx)+isin(nx) so:

    de Moivre - (cosx+isinx)^5=cos(5x)+isin(5x) and
    binomial - (cosx+isinx)^5=cos^5(x)+5cos^4(x)isin(x)-10cos^3(x)sin^2(x)-10cos^2(x)isin^3(x)+5cos(x)sin^4(x)+sin^5(x)i

    So,
    (1) sin5x=5cos^4(x)sin(x)-10cos^2(x)sin^3(x)+sin^5(x)

    After some arithmetic arrangements and and Pythagorean trigonometric identity I got this: sin5x=12sin^5(x)-12sin^3(x)+sin(x) and task under a) is completed (I think).

    But the problem is at task b), so, I don't know how to write expression (1) in terms of cosx function. Is there any identity? Or maybe some other method.

    (Second problem) If z is a complex number, what does this equation represent: |z|-2Re(z)=4

    Attempt: No attempt. I need a complete solution and explanation.

    Thank you!
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  2. #2
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    Quote Originally Posted by patzer View Post
    Hello!

    (First problem) Express sin5x in terms of a) sinx and b) cosx

    Attempt: According to de Moivre's formula - (cosx+isinx)^n=cos(nx)+isin(nx) so:

    de Moivre - (cosx+isinx)^5=cos(5x)+isin(5x) and
    binomial - (cosx+isinx)^5=cos^5(x)+5cos^4(x)isin(x)-10cos^3(x)sin^2(x)-10cos^2(x)isin^3(x)+5cos(x)sin^4(x)+sin^5(x)i

    So,
    (1) sin5x=5cos^4(x)sin(x)-10cos^2(x)sin^3(x)+sin^5(x)

    After some arithmetic arrangements and and Pythagorean trigonometric identity I got this: sin5x=12sin^5(x)-12sin^3(x)+sin(x) and task under a) is completed (I think).

    But the problem is at task b), so, I don't know how to write expression (1) in terms of cosx function. Is there any identity? Or maybe some other method.

    (Second problem) If z is a complex number, what does this equation represent: |z|-2Re(z)=4

    Attempt: No attempt. I need a complete solution and explanation.

    Thank you!

    You'll get a complete hint and, perhaps, some explanation. The rest is on you.

    Write z=x+iy\,,\,\,x,y\in\mathbb{R} , so

    |z|-2Re(z)=4\Longleftrightarrow \sqrt{x^2+y^2}-2x=4\Longrightarrow x^2+y^2=4x^2-16x+16\Longrightarrow

     3\left(x-\frac{8}{3}\right)^2-\frac{64}{3}-y^2=-16 ...and now remember a little analytic geometry and you're done.

    Tonio
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