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 real-class 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 z6 using De Moivre's Theorem,
which will yield z6 = -8i.
E.g. Write (3
+ i)7 in the form x + iy.
Solution : First determine the radius,
r = |3
+ i|
r =
(x2
+ y2)
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 De-Moivre's theorem,
= [27(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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.
2 (cos
/4 + i sin
3
+ i)7 in the form x + iy.
=
+ i sin 30