This is a free lesson from our course in Trigonometry
In this lesson you'll explore how to use De Moivre's Theorem; with the help of several examples with
solution and using video explanations by the instructor that brings in an element
of realclass room experience. In addition, there are practice questions and quizzes
with solution to strengthen learning. To change a complex number into standard form a + bi using
De Moivre's Theorem, you need to change the complex number into polar form. For example, to write (1 + i)^{6}
in standard form, consider z = 1 + i and convert it to polar form as you
have learned earlier. Compare it with z = x + iy and
then substitute the values of x and y in terms of r and .
(More text below video...)
(Continued from above)So the complex number
in the polar form will be z =
2 (cos
/4 + i sin
/4). Now determine (1
+ i)^{6}, to work out z^{6} using De Moivre's Theorem,
which will yield z^{6} = 8i.
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
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