Polar Form Representation
In Further Mathematics, complex numbers are expressed in **polar form** $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ is the modulus (distance from the origin) and $\theta = \arg(z)$ is the argument (angle with the positive real axis).
De Moivre's Theorem Definition
De Moivre's Theorem simplifies raising complex numbers to a power $n$. Instead of expanding brackets, we raise the modulus to the power of $n$ and multiply the argument by $n$:
Key Applications of the Theorem
You must master the two main applications of De Moivre's Theorem in further pure exams:
Expanding Trig Functions ($\cos n\theta$ and $\sin n\theta$)
By expanding $(\cos\theta + i\sin\theta)^n$ using Binomial Expansion, and equating the real and imaginary parts to $\cos(n\theta) + i\sin(n\theta)$, you can express multiple-angle trig functions in powers of single-angle trig functions.
Finding Complex Roots ($z^n = w$)
To find the $n$-th roots of a complex number, express $w$ in polar form, add $2k\pi$ to the argument, apply the fractional power $1/n$ using De Moivre's Theorem, and solve for $k = 0, 1, ..., n-1$. The roots will form a regular polygon in the Argand diagram.
Worked Exam Example (Trig Expansion)
Worked Further Maths Example
Question: Express $\cos(3\theta)$ in terms of powers of $\cos\theta$.
Solution:
- State De Moivre's Theorem for $n=3$: $$\cos(3\theta) + i\sin(3\theta) = (\cos\theta + i\sin\theta)^3$$
- Expand the right-hand side using binomial expansion: $$(\cos\theta + i\sin\theta)^3 = \cos^3\theta + 3\cos^2\theta(i\sin\theta) + 3\cos\theta(i\sin\theta)^2 + (i\sin\theta)^3$$
- Evaluate powers of $i$ (remember $i^2 = -1$ and $i^3 = -i$): $$= \cos^3\theta + 3i\cos^2\theta\sin\theta - 3\cos\theta\sin^2\theta - i\sin^3\theta$$
- Group real and imaginary parts: $$= (\cos^3\theta - 3\cos\theta\sin^2\theta) + i(3\cos^2\theta\sin\theta - \sin^3\theta)$$
- Equate the real parts on both sides of the equation: $$\cos(3\theta) = \cos^3\theta - 3\cos\theta\sin^2\theta$$
- Substitute $\sin^2\theta = 1 - \cos^2\theta$ to get the expression entirely in terms of $\cos\theta$: $$\cos(3\theta) = \cos^3\theta - 3\cos\theta(1 - \cos^2\theta)$$ $$\cos(3\theta) = \cos^3\theta - 3\cos\theta + 3\cos^3\theta$$ $$\cos(3\theta) = 4\cos^3\theta - 3\cos\theta$$
Result: $\cos(3\theta) = 4\cos^3\theta - 3\cos\theta$.
Frequently Asked Questions
Euler's relation states that $e^{i\theta} = \cos\theta + i\sin\theta$. Therefore, any complex number can be written as $z = re^{i\theta}$, which is extremely convenient for multiplication and division calculations.
The $n$-th roots of unity are the solutions to the equation $z^n = 1$. They lie on a unit circle ($r=1$) in the Argand diagram and are equally spaced by an angle of $2\pi/n$ radians.
Master A-Level Further Mathematics
Work 1-on-1 with Oxford and Cambridge Further Maths graduates to master complex numbers, matrices, and differential equations. Book your free consultation today.