De Moivre’s Theorem & simple applications

De Moivre’s Theorem & simple applications By Chtan FYHS-Kulai

In mathematics, de Moivre‘s formula, named after Abraham de Moivre. By Chtan FYHS-Kulai

The formula is important because it connects complex numbersand trigonometry. The expression "cos x + i sin x" is sometimes abbreviated to "cis x". By Chtan FYHS-Kulai

By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x). By Chtan FYHS-Kulai

Furthermore, one can use a generalization of this formula to find explicit expressions for the n-th roots of unity, that is, complex numbers z such that zn = 1. By Chtan FYHS-Kulai

De Moivre’s theorem For all values of n, the value, or one of the values in the case where n is fractional, of is By Chtan FYHS-Kulai

Proofing of De Moivre’s Theorem By Chtan FYHS-Kulai

Now, let us prove this important theorem in 3 parts. • When n is a positive integer • When n is a negative integer • When n is a fraction By Chtan FYHS-Kulai

Case 1 : if n is a positive integer By Chtan FYHS-Kulai

By Chtan FYHS-Kulai

Continuing this process, when n is a positive integer, By Chtan FYHS-Kulai

Case 2 : if n is a negative integer Let n=-m where m is positive integer By Chtan FYHS-Kulai

Case 3 : if n is a fraction equal to p/q, p and q are integers By Chtan FYHS-Kulai

Raising the RHS to power q we have, but, By Chtan FYHS-Kulai

Hence, De Moivre’s Theorem applies when n is a rational fraction. By Chtan FYHS-Kulai

Proofing by mathematical induction By Chtan FYHS-Kulai

The hypothesis of Mathematical Induction has been satisfied , and we can conclude that By Chtan FYHS-Kulai

e.g. 1 Let z = 1 − i. Find. Soln: First write z in polar form. By Chtan FYHS-Kulai

Polar form : Applying de Moivre’s Theorem gives : By Chtan FYHS-Kulai

It can be verified directly that By Chtan FYHS-Kulai

Properties of By Chtan FYHS-Kulai

If then By Chtan FYHS-Kulai

Hence, By Chtan FYHS-Kulai

Similarly, if Hence, By Chtan FYHS-Kulai

We have, Maximum value of cosθ is 1, minimum value is -1. Hence, normally By Chtan FYHS-Kulai

What happen, if the value of is more than 2 or less than -2 ? By Chtan FYHS-Kulai

e.g. 2 Given that Prove that By Chtan FYHS-Kulai

e.g. 3 If , find By Chtan FYHS-Kulai

Do take note of the following : By Chtan FYHS-Kulai

e.g. 4 By Chtan FYHS-Kulai

Applications of De Moivre’s theorem By Chtan FYHS-Kulai

We will consider three applications of De Moivre’s Theoremin this chapter. 1. Expansion of . 2. Values of . 3. Expressions for in terms of multiple angles. By Chtan FYHS-Kulai

Certain trig identities can be derived using De Moivre’s theorem. In particular, expression such as can be expressed in terms of : By Chtan FYHS-Kulai

e.g. 5 Use De Moivre’s Thorem to find an identity for in terms of . By Chtan FYHS-Kulai

e.g. 6 Find all complex cube roots of 27i. Soln: We are looking for complex number z with the property Strategy : First we write 27i in polar form :- By Chtan FYHS-Kulai

Now suppose Satisfies . Then, by De Moivre’s Theorem, By Chtan FYHS-Kulai

This means : where k is an integer. Possibilities are : k=0, k=1, k=2 By Chtan FYHS-Kulai

In general : to find the complex nth roots of a non-zero complex number z. 1. Write z in polar form : By Chtan FYHS-Kulai

2. z will have n different nth roots (i.e. 3 cube roots, 4 fourth roots, etc.) 3. All these roots will have the same modulus the positive real nth roots of r) . 4. They will have different arguments : By Chtan FYHS-Kulai

5. The complex nth roots of z are given (in polar form) by …etc By Chtan FYHS-Kulai

e.g. 7 Find all the complex fourth roots of -16. Soln: Modulus = 16 Argument = ∏ By Chtan FYHS-Kulai

Fourth roots of 16 all have modulus : and possibilities for the arguments are : By Chtan FYHS-Kulai

Hence, fourth roots of -16 are : By Chtan FYHS-Kulai

e.g. 8 Given that and find the value of m. By Chtan FYHS-Kulai

De Moivre’s Theorem & simple applications