9231. FP2. Complex Numbers

From earlier work you should already know the following:

Complex numbers in rectangular form: x+yi;
Complex numbers in polar form (trigonometric & exponential);
Arithmetic with complex numbers (remember that multiplication in polar involves multiplying the moduli and adding the arguments);
Fundamental Theorem of Algebra: A degree n equation has n roots, all either real or in complex conjugate pairs.

Demoivre’s Theorem

If z=cos𝜽+isin𝜽, then zⁿ=cos(n𝜽) + isin(n𝜽)

This can be proved using induction.

We can use DeMoivre’s theorem to calculate cos(n𝜽) or sin(n𝜽) as a sum of powers of cos𝜽 or sin𝜽, by using the Binomial Theorem to expand zⁿ, using Pythagoras’ Theorem to change to just one trigonometric function, and then comparing real and imaginary parts.

This can also be used to solve certain equations in x by replacing x with a trigonometric function.

Worked Example 1 – Using DeMoivre’s Theorem

Find the value of:

(a) $(cos \frac{ \pi}{4} + is in \frac{ \pi}{4} )^8$

(b) $\frac{1}{ (cos( - \frac{ 3 \pi }{4} ) + i sin ( - \frac{ 3 \pi}{4} ))^6}$

(d) $( \sqrt{3} - i)^{18}$

Worked Example 2 – Expressing Trigonometric Functions of Multiple Angles

Express cos5𝜽 in terms of cos𝜽.

Express sin3𝜽 in terms of sin𝜽.

Express tan4𝜽 in terms of tan𝜽.

Worked Example 3 – Solving Polynomial Equations using Substitution

By considering the form of tan3𝜽, solve the cubic equation $3t^3 + 6t^2 -9t -2 = 0$

Exercise 1

Answers

Exercise 1 Worked Solutions

Further Uses of DeMoivre’s Theorem

We can use Demoivre’s theorem to write $z+ \frac{1}{z}$ or $z- \frac{1}{z}$ as purely real or purely imaginary.

We can do the same for zⁿ+z^-n or zⁿ-z^-n.

By comparing this with the binomial expansion of these expressions, we can rewrite trigonometric functions raised to a power as trigonometric functions of multiple angles, allowing us to integrate them.

N.B. The above can also be usefully written in exponential form.

Worked Example

Evaluate $\int sin^6 \theta d \theta$

Express cos⁶𝜽 – sin⁶𝜽 in the form pcosq𝜽 + rcoss𝜽, where p, q, r and s are constants to be determined.

In summary, for your reference:

Exercise 2

Answers

Exercise 2 Worked Solutions

Roots of Unity

Consider z³=1. Its roots can be found in cartesian form by dividing through by the factor z-1 (as 1 is obviously a root).

They can also be found using DeMoivre’s theorem, by rewriting z as (cos(0+2π)+isin(0+2π))^1/3.

If we draw the roots on an Argand diagram we see they are equally spaced, so give a regular shape. We also note from (i) above that they sum to zero (which we can also see by representing them as vectors).

N.B. We can do this with any complex number, not just 1.

Worked Example

Find, in exponential form, the 7th roots of unity.

Solve the equation z⁶=1. Show the roots on an Argand diagram.
One fourth root of the complex number w is 2+3i. Find w and its other fourth roots and represent all fourth roots on an Argand diagram.

Exercise 3

Answers

Exercise 3 Worked Solutions

More advanced problems, including series

Our formulae for z+z^-1 and z-z^-1can often help us when tackling problems, by making it easier to write an expression with its real and imaginary parts separated.

Please make careful note that we are now solving purely trigonometric problems, by working with the complex number and then considering real and imaginary parts. This is often the easiest method.