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

(Continued from above)For example, if z = r (cos + i sin ) is a complex number, then
zn = rn[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 = (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

De Moivre's theorem can be extended to roots of complex numbers yielding the nth root theorem.

Winpossible's online math courses and tutorials have gained rapidly popularity since their launch in 2008. Over 100,000 students have benefited from Winpossible's courses... these courses in conjunction with free unlimited homework help serve as a very effective math-tutor for our students.
 - All of the Winpossible math tutorials have been designed by top-notch instructors and offer a comprehensive and rigorous math review of that topic. - We guarantee that any student who studies with Winpossible, will get a firm grasp of the associated problem-solving techniques. Each course has our instructors providing step-by-step solutions to a wide variety of problems, completely demystifying the problem-solving process! - Winpossible courses have been used by students for help with homework and by homeschoolers. - Several teachers use Winpossible courses at schools as a supplement for in-class instruction. They also use our course structure to develop course worksheets.

 Copyright © Winpossible, 2010 - 2011 Best viewed in 1024x768 & IE 5.0 or later version