Trigonometric Form of Complex Numbers

Key Questions

  • What is DeMoivre's theorem?

    DeMoivre's Theorem expand's on Euler's formula:
    e^(ix)=cosx+isinx

    DeMoivre's Theorem says that:

    • (e^(ix))^n=(cosx+isinx)^n
    • (e^(ix))^n=e^(i nx)
    • e^(i nx)=cos(nx)+isin(nx)
    • cos(nx)+isin(nx)-=(cosx+isinx)^n

    Example:
    cos(2x)+isin(2x)-=(cosx+isinx)^2

    (cosx+isinx)^2=cos^2x+2icosxsinx+i^2sin^2x

    However, i^2=-1

    (cosx+isinx)^2=cos^2x+2icosxsinx-sin^2x

    Resolving for real and imaginary parts of x:
    cos^2x-sin^2x+i(2cosxsinx)

    Comparing to cos(2x)+isin(2x)

    cos(2x)=cos^2x-sin^2x
    sin(2x)=2sinxcosx
    These are the double angle formulae for cos and sin

    This allows us to expand cos(nx) or sin(nx) in terms of powers of sinx and cosx

    DeMoivre's theorem can be taken further:
    Given z=cosx+isinx

    z^n=cos(nx)+isin(nx)

    z^(-n)=(cosx+isinx)^(-n)=1/(cos(nx)+isin(nx))

    z^(-n)=1/(cos(nx)+isin(nx))xx(cos(nx)-isin(nx))/(cos(nx)-isin(nx))=(cos(nx)-isin(nx))/(cos^2(nx)+sin^2(nx))=cos(nx)-isin(nx)

    z^n+z^(-n)=2cos(nx)
    z^n-z^(-n)=2isin(nx)

    So, if you wanted to express sin^nx in terms of multiple angles of sinx and cosx:
    (2isinx)^n=(z-1/z)^n

    Expand and simply, then input values for z^n+z^(-n) and z^n-z^(-n) where necessary.

    However, if it involved cos^nx, then you would do (2cosx)^n=(z+1/z)^n and follow the similar steps.

    1s2s2p · 1 · Mar 5 2018
  • Why do you need to find the trigonometric form of a complex number?

    Depending on what you need to do with your complex numbers, the trigonometric form can be very useful or very thorny.

    For example, let z_1=1+i, z_2=sqrt(3)+i and z_3=-1+i sqrt{3}.
    Let's compute the two trigonometric forms:
    theta_1=arctan(1)=pi/4 and rho_1=sqrt{1+1}=sqrt{2}
    theta_2=arctan(1/sqrt{3})=pi/6 and rho_2=sqrt{3+1}=2
    theta_3=pi + arctan(-sqrt{3})=2/3 pi and rho_3=sqrt{1+3}=2
    So the trigonometric forms are:
    z_1=sqrt{2}(cos(pi/4)+i sin(pi/4))
    z_2=2(cos(pi/6)+i sin(pi/6))
    z_3=2(cos(2/3 pi)+i sin(2/3 pi))

    Addition
    Let's say you want to compute z_1+z_2+z_3. If you use the algebraic form, you get
    z_1+z_2+z_3=(1+i)+(sqrt{3}+i)+(-1+i sqrt{3})=sqrt{3}+i(2+sqrt{3})
    Quite easy. Now try with the trigonometric form...
    z_1+z_2+z_3=sqrt{2}(cos(pi/4)+i sin(pi/4))+2(cos(pi/6)+i sin(pi/6))+2(cos(2/3 pi)+i sin(2/3 pi))
    it turns out that the shortest way to add these two expressions is to solve cosines and sines, which means... turning to the algebraic form!
    The algebraic form is often the best form to choose in adding complex numbers.

    Multiplication
    Now we try to compute z_1*z_2*z_3. Using algebraic forms requires a lot of annoying computations. But solving this product with the trigonometric forms is simpler:
    z_1*z_2*z_3=sqrt{2}(cos(pi/4)+i sin(pi/4))*2(cos(pi/6)+i sin(pi/6))*2(cos(2/3 pi)+i sin(2/3 pi))=4 sqrt{2}(cos(pi/4+pi/6+2/3 pi)+i sin(pi/4+pi/6+2/3 pi))=4 sqrt{2}(cos(13/12 pi)+i sin(13/12 pi))
    The ingredients to prove that the second equality holds come from trigonometry: the two addition formulae
    sin(alpha + beta)=sin(alpha)cos(beta)+sin(beta)cos(alpha)
    cos(alpha + beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta)
    Multiplication of complex numbers is even cleaner (but conceptually not easier) in exponential form.

    In some sense, the trigonometric form is a sort of in-between form between the algebraic and the exponential forms. The trigonometric form is the way to switch between these two. In this sense it's a sort of a "dictionary" to "translate" forms.

    Maurizio Giaffredo · 1 · Jan 8 2015
  • How do you find a trigonometric form of a complex number?

    Let z=x+iy a complex number in algebraic form.

    z=r(cosphi+isinphi) is its trigonometric form, where:

    r=sqrt(x^2+y^2) is the modulus of the number and

    • if x>0

    phi=arctan(y/x) ,

    • if x<0

    phi=arctan(y/x)+pi,

    • if x=0 and y>0

    phi=pi/2,

    • if x=0 and y<0

    phi=3/2pi

    • if x=y=0

    It's all zero!

    Massimiliano · 1 · Feb 22 2015

Questions

Complex Numbers in Trigonometric Form