De Moivre’s theorem and roots of unity – further maths worksheet with answers to download as PDF

Use de Moivre’s theorem, \((\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta\), to evaluate each expression. Give exact answers in the form \(a+bi\).

1. \(\left(\cos\tfrac\pi6+i\sin\tfrac\pi6\right)^5\)
2. \((1+i)^8\)
3. \((\sqrt3-i)^6\)
4. \((-1+i)^5\)

Euler’s formula states that \(e^{i\theta}=\cos\theta+i\sin\theta\).

1. Write \(3e^{i\pi/3}\) in the form \(a+bi\).
2. Write \(1-i\) in the form \(re^{i\theta}\).
3. Simplify \(2e^{i\pi/4}\times3e^{i\pi/6}\) and \(\dfrac{6e^{i\pi/2}}{2e^{i\pi/3}}\).
4. Show that \(\left(e^{i\pi/6}\right)^{12}=1\) and that \(e^{i\pi}+1=0\).

Let \(c=\cos\theta\) and \(s=\sin\theta\).

1. Expand \((c+is)^3\) using the binomial theorem.
2. By equating real parts, show that \(\cos3\theta=4\cos^3\theta-3\cos\theta\).
3. By equating imaginary parts, show that \(\sin3\theta=3\sin\theta-4\sin^3\theta\).
4. Check both results when \(\theta=\tfrac\pi6\).

Let \(z=\cos\theta+i\sin\theta\), so that \(z+\dfrac1z=2\cos\theta\) and \(z-\dfrac1z=2i\sin\theta\), and \(z^n+\dfrac1{z^n}=2\cos n\theta\).

1. Expand \(\left(z+\dfrac1z\right)^3\) and hence show that \(\cos^3\theta=\tfrac14(\cos3\theta+3\cos\theta)\).
2. Expand \(\left(z-\dfrac1z\right)^4\) and hence show that \(\sin^4\theta=\tfrac18(\cos4\theta-4\cos2\theta+3)\).
3. Use part (b) to evaluate \(\displaystyle\int_0^{\pi/2}\sin^4\theta\,d\theta\).

Solve each equation. Give the roots in Cartesian form with exact values and show them on an Argand diagram.

1. \(z^3=8\)
2. \(z^3=-8i\)
3. State the sum of the three roots in each case.

The sixth roots of unity are the solutions of \(z^6=1\). Let \(\omega=e^{i\pi/3}\).

1. Write down all six roots in the form \(\omega^k\) and in Cartesian form.
2. Show that \(\omega^2=\omega-1\) and \(\omega^3=-1\).
3. Show that the sum of the six roots is zero.
4. Find the product of the six roots.

A marker on the rim of a gear wheel of radius 5 cm is at the point \(z=5e^{3it}\) in an Argand diagram whose origin is the centre of the wheel. Here \(t\) is the time in seconds and the angle is in radians.

1. Find the position of the marker when \(t=2\), in the form \(a+bi\) to 2 decimal places.
2. How long does the wheel take to make one complete turn?
3. The velocity of the marker is \(\dfrac{dz}{dt}\). Find it and show that the speed of the marker is constant. Find that speed.
4. Show that the velocity is always perpendicular to the position, by writing \(\dfrac{dz}{dt}=3i\,z\).

A garden designer places five sprinklers at the corners of a regular pentagon centred on the fountain. In an Argand diagram with the fountain at the origin and distances in metres, the sprinklers are at the solutions of \(z^5=1024\).

1. Find the modulus of each sprinkler position and write down all five positions in modulus-argument form.
2. Calculate the distance between two neighbouring sprinklers to 2 decimal places.
3. A path runs along all five sides. Find its total length.
4. Find the area enclosed by the path, using five triangles with their apex at the fountain.

A tile designer draws a regular hexagon whose six corners are the roots of \(z^6=-64\) in an Argand diagram, with units in decimetres.

1. Find the modulus and argument of each root.
2. Write the roots in the form \(a+bi\).
3. What is the length of one side of the hexagon?
4. Hence find the area of one hexagonal tile to 3 significant figures.

The intensity of light passing through a polarising filter varies as \(I=100\cos^4\omega t\) watts per square metre, where \(\omega\) is a constant and \(t\) is time.

1. Use \(\left(z+\dfrac1z\right)^4\) with \(z=\cos\omega t+i\sin\omega t\) to show that \(\cos^4\theta=\tfrac18(\cos4\theta+4\cos2\theta+3)\).
2. Hence write \(I\) as a sum of cosines of multiples of \(\omega t\).
3. Find the mean intensity over one period.

Four atoms in a square crystal cell are located at the solutions of \(z^4=-16\), where distances are in nanometres.

1. Find the four positions in the form \(a+bi\).
2. Show that the four atoms form a square and find its side length.
3. A fifth atom sits at the origin. Find the distance from it to each of the others.

The British 50p coin is a regular heptagon (seven sides). Model a coin by placing its vertices at the seven solutions of \(z^7=14^7\), with distances in millimetres.

1. Find the modulus and write the seven vertices in exponential form.
2. Show that the vertices have sum zero.
3. Find the length of one side to 2 decimal places.
4. Find the length of the longest diagonal joining two vertices, to 2 decimal places.

A faulty amplifier turns an input signal \(x=2\sin\theta\) volts into the output \(y=x^3\), where \(\theta=\omega t\). Sound engineers call the extra frequencies in the output distortion.

1. Use \(\sin3\theta=3\sin\theta-4\sin^3\theta\) to show that \(\sin^3\theta=\tfrac14(3\sin\theta-\sin3\theta)\).
2. Write the output \(y\) as a sum of sines of \(\theta\) and \(3\theta\).
3. Find the amplitude of the third harmonic \(\sin3\theta\) as a percentage of the amplitude of the fundamental \(\sin\theta\).

A giant observation wheel has radius 60 m and its centre is 67.5 m above the ground. It has 32 identical capsules equally spaced round the rim. Let the centre be the origin of an Argand diagram, and let capsule \(k\) be at \(60e^{i\phi_k}\) where \(\phi_k=-\tfrac\pi2+\tfrac{2\pi k}{32}\) at a certain moment, for \(k=0,1,\dots,31\).

1. Show that capsule 0 is at the bottom of the wheel and find its height above the ground.
2. Find the argument of the capsule at the top and its height above the ground.
3. Find the height of capsule 8.
4. Find the sum of the 32 complex positions, and explain the result.

In this problem you find an exact value of \(\cos\tfrac\pi{10}\).

1. Expand \((\cos\theta+i\sin\theta)^5\) and show that \(\cos5\theta=16\cos^5\theta-20\cos^3\theta+5\cos\theta\).
2. Explain why \(\cos5\theta=0\) when \(\theta=\tfrac\pi{10}\) and when \(\theta=\tfrac{3\pi}{10}\).
3. Deduce that \(\cos^2\tfrac\pi{10}\) is a root of \(16x^2-20x+5=0\).
4. Hence find the exact value of \(\cos^2\tfrac\pi{10}\), and show that \(\cos\tfrac\pi{10}=\sqrt{\dfrac{5+\sqrt5}8}\).

Let \(w=-8+8\sqrt3\,i\).

1. Express \(w\) in modulus-argument form.
2. Find the four fourth roots of \(w\) in the form \(r(\cos\theta+i\sin\theta)\) and then in the form \(a+bi\).
3. Show the roots on an Argand diagram and prove that they form a square.
4. Find the area of the square.

Let \(C=1+\cos\theta+\cos2\theta+\dots+\cos(n-1)\theta\) and \(S=\sin\theta+\sin2\theta+\dots+\sin(n-1)\theta\), and let \(z=\cos\theta+i\sin\theta\).

1. Explain why \(C+iS=1+z+z^2+\dots+z^{n-1}\).
2. Sum the geometric series and show that \(C+iS=\dfrac{1-z^n}{1-z}\).
3. By writing \(1-z^n=-2i\sin\tfrac{n\theta}2\,e^{in\theta/2}\) and similarly for \(1-z\), show that \(C=\dfrac{\sin\tfrac{n\theta}2\cos\tfrac{(n-1)\theta}2}{\sin\tfrac\theta2}\).
4. Use the formula to find \(C\) when \(\theta=\tfrac\pi4\) and \(n=7\), and check by adding the seven terms.

Let \(\omega=\cos\tfrac{2\pi}5+i\sin\tfrac{2\pi}5\) be a fifth root of unity.

1. Explain why \(1+\omega+\omega^2+\omega^3+\omega^4=0\).
2. Let \(a=\omega+\omega^4\) and \(b=\omega^2+\omega^3\). Show that \(a=2\cos72^\circ\) and \(b=2\cos144^\circ\), and that \(a+b=-1\).
3. Show that \(ab=-1\).
4. Hence find the exact value of \(\cos72^\circ\).

Let \(f(z)=z^6-1\).

1. Write down the six roots of \(f(z)=0\) in modulus-argument form.
2. Pair each non-real root with its conjugate and show that \(f(z)=(z-1)(z+1)(z^2-z+1)(z^2+z+1)\).
3. Show that \(z^2+z+1=0\) has the roots \(e^{\pm2\pi i/3}\) and check your factorisation by expanding.
4. Hence prove that \(\cos\tfrac\pi3=\tfrac12\) using the sum of two conjugate roots.

Consider the equation \((z+1)^5=z^5\).

1. Explain why \(z=0\) is not a root and why the equation reduces to a polynomial of degree 4.
2. Put \(w=\dfrac{z+1}z\). Show that \(w^5=1\) and \(w\ne1\).
3. Hence show that \(z=\dfrac1{w-1}\) and find \(z\) in terms of \(\theta=\tfrac{2\pi k}5\) for \(k=1,2,3,4\).
4. Show that \(\dfrac1{e^{i\theta}-1}=-\dfrac12-\dfrac i2\cot\tfrac\theta2\). Hence write all four roots and check one numerically.