\left[r(\cos\theta+i\sin\theta)\right]^n?

r^n(\cos n\theta+i\sin n\theta)

How is De Moivre's theorem proved for positive integers?

By induction, using the addition formulae for sine and cosine.

How is it extended to negative integers?

(\cos\theta+i\sin\theta)^{-m}=\frac{1}{\cos m\theta+i\sin m\theta}=\cos m\theta-i\sin m\theta.

How do you find \cos n\theta in powers of \cos\theta and \sin\theta?

Expand (c+is)^n and take the real part.

\cos5\theta in terms of \cos\theta?

16\cos^5\theta-20\cos^3\theta+5\cos\theta

How do you write \sin^4\theta in multiple angles?

Expand \left(z-\frac1z\right)^4=16\sin^4\theta and pair z^k+z^{-k} terms.

\sin^4\theta in multiple angles?

\frac18(\cos4\theta-4\cos2\theta+3)

De Moivre's theorem and trigonometric identitiesEdexcel International A Level Further Maths: Flashcards

Card 1 of 130 of 13 known

Question

State De Moivre's theorem.

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

State De Moivre's theorem.: $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$ for any integer $n$ .
$\left[r(\cos\theta+i\sin\theta)\right]^n$ ?: $r^n(\cos n\theta+i\sin n\theta)$
How is De Moivre's theorem proved for positive integers?: By induction, using the addition formulae for sine and cosine.
How is it extended to negative integers?: $(\cos\theta+i\sin\theta)^{-m}=\frac{1}{\cos m\theta+i\sin m\theta}=\cos m\theta-i\sin m\theta$ .
$z^n+z^{-n}$ when $z=\cos\theta+i\sin\theta$ ?: $2\cos n\theta$
$z^n-z^{-n}$ when $z=\cos\theta+i\sin\theta$ ?: $2i\sin n\theta$
How do you find $\cos n\theta$ in powers of $\cos\theta$ and $\sin\theta$ ?: Expand $(c+is)^n$ and take the real part.
$\cos3\theta$ in terms of $\cos\theta$ ?: $4\cos^3\theta-3\cos\theta$
$\sin3\theta$ in terms of $\sin\theta$ ?: $3\sin\theta-4\sin^3\theta$
$\cos5\theta$ in terms of $\cos\theta$ ?: $16\cos^5\theta-20\cos^3\theta+5\cos\theta$
How do you write $\sin^4\theta$ in multiple angles?: Expand $\left(z-\frac1z\right)^4=16\sin^4\theta$ and pair $z^k+z^{-k}$ terms.
$\sin^4\theta$ in multiple angles?: $\frac18(\cos4\theta-4\cos2\theta+3)$
$\int_0^{\pi/2}\sin^4\theta\,d\theta$ ?: $\frac{3\pi}{16}$

Exam questions on De Moivre's theorem and trigonometric identities

A complex number is $z=\cos\theta+i\sin\theta$ , where $\theta$ is real.
Given that $\theta=\frac\pi9$ , find $z^6$ in the form $a+ib$ , giving exact values of $a$ and $b$ .2 marks
Let $c=\cos\theta$ and $s=\sin\theta$ , and consider $(c+is)^3$ expanded using the binomial theorem.
Hence express $\cos3\theta$ in terms of $\cos\theta$ only.2 marks
De Moivre's theorem states that $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$ for every integer $n$ . A student proves the case $n\ge1$ by induction.
Show that if the result is true for $n=k$ , where $k\ge1$ , then it is true for $n=k+1$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).