Applications of de Moivre's Theorem (Edexcel A-Level Further Mathematics): Revision Notes

Example: Expanding $(\cos \theta + i \sin \theta)^2$

Applying de Moivre's Theorem with$( n = 2 )$:

Expanding the left-hand side algebraically:

By comparing real and imaginary parts, we obtain the well-known double-angle identities:

• $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$
• $\sin(2\theta) = 2 \cos \theta \sin \theta$

Example: Solving $(\cos \theta + i \sin \theta)^3 = 1$

We need to find the values of $\theta$ for which:

Using de Moivre's Theorem:

This implies that $3\theta = 2k\pi$ for some integer $k$, since the argument of $1 + 0i$ is $0$.

Solving for $\theta$

Thus, the solutions are:

Example: Finding the $4th$ Roots of $16$

We know that $16 = 16 (\cos 0 + i \sin 0)$ is a complex number in modulus-argument form.

Using de Moivre's Theorem, the $4th$ roots of $16$ are given by:

For $k = 0, 1, 2, 3$, the four roots are:

• $z_0 = 2 (\cos 0 + i \sin 0) = 2$
• $z_1 = 2 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) = 2i$
• $z_2 = 2 \left( \cos \pi + i \sin \pi \right) = -2$
• $z_3 = 2 \left( \cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2} \right) = -2i$ Thus, the $4th$ roots of $16$ are 2, 2i, -2, -2i

Example: Deriving the Triple Angle Formula

We can use de Moivre's Theorem to derive the triple angle formula for sine and cosine.

Using $n=3$, we have:

Expanding the left-hand side:

By comparing real and imaginary parts, we obtain:

• $\cos(3 \theta) = \cos^3 \theta - 3 \cos \theta \sin^2 \theta$
• $\sin(3 \theta) = 3 \cos^2 \theta \sin \theta - \sin^3 \theta$ These are the triple-angle formulas for sine and cosine.

Example: Summing a Cosine Sequence Consider the sequence:

We want to find a way to simplify this sum. To do this, we can use Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, and de Moivre's Theorem.

Now, let's also consider the sequence of sine terms:

We can combine these two sequences into a complex number:

Using Euler's formula, this becomes:

This is a geometric series with:

• First term $a=1$
• Common ratio $r = e^{i\theta}$
• Number of terms $n+1$

The sum of a geometric series is given by:

Substituting our values:

Step 1: Simplifying the Denominator

To simplify the denominator, recall that $e^{i\theta} = \cos \theta + i \sin \theta$.

Therefore:

Now, multiply the numerator and denominator by the conjugate of the denominator to simplify it further:

This gives:

Step 2: Simplifying the Numerator

Next, expand and simplify the numerator. After expanding, we focus on the real part of the expression to find the sum for $C$, because we are interested in summing cosine terms.

After some algebra, we get:

Since $\cos^2 \theta + \sin^2 \theta = 1$, this simplifies to:

Further simplifying:

This is the simplified sum for the cosine sequence.