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...)
Other useful lessons:
 Getting Started: De Moivre's Theorem Roots of Complex Numbers

(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 = �, 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)
= -64 3 - 64i

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