This is a free lesson from our course in Trigonometry
This lesson content walks you through the De Moivre's Theorem. The trigonometric and exponential formulation is explained introducing
the complex number definition in standard form. The simplification
division of complex numbers is performed with the use of exponential forms. The
formula is important because it connects complex number and trigonometry. The special
feature is that you'll find all this with video explanation by instructor in own
handwriting, some examples and practice questions with solution. According to the theorem; a complex number raised to a given positive integral
power is equal to the modulus of the number raised to the power multiplied by the
amplitude times the given power.
(More text below video...)
(Continued from above)For example, if z = r (cos
+ i sin
) is a complex number,
then z^{n}
= r^{n}[cos (n)
+ sin (n)],
where n is an integer.
The expression "cos
+ i sin
" is sometimes abbreviated
to "cis
".
Problems involving powers of complex numbers can be solved using binomial expansion,
but applying De Moivre's theorem is usually more direct.
E.g. Write (3
+ i)^{7} in the form x + iy.
Solution : First determine the radius,
r = 3
+ i
r =
(x^{2}
+ y^{2})
r =
(3
+ 1)
r = 2
Since cos
=
3/2
and sin
= ½,
must be in the first quadrant and
= 30°. Therefore,
(3
+ i)^{7} = [2(cos 30
+ i sin 30)]^{7}
Using DeMoivre's theorem,
= [2^{7}(cos 7(30)
+ i sin 7 (30))]
= 128(cos 210
+ i sin 210)
= 128 ((3/2)(1/2)i)
= 643
 64i
De Moivre's theorem can be extended to roots of complex numbers yielding the nth
root theorem.
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