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

(a) Express \(1-i\) in modulus-argument form. [2]

(b) Use de Moivre’s theorem to show that \((1-i)^{10}=-32i\). [2]

(a) By expanding \(\left(z-\dfrac1z\right)^4\) where \(z=\cos\theta+i\sin\theta\), show that \(\sin^4\theta=\tfrac18(\cos4\theta-4\cos2\theta+3)\). [3]

(b) Hence find the exact value of \(\displaystyle\int_0^{\pi/4}\sin^4\theta\,d\theta\). [2]

(a) Solve \(z^3=-27i\), giving the roots in the form \(r(\cos\theta+i\sin\theta)\) with \(-\pi\lt \theta\le\pi\). [3]

(b) Show the roots on an Argand diagram and find the area of the triangle they form. [2]

Using \(\cos3\theta=4\cos^3\theta-3\cos\theta\), solve \(\cos3\theta=\cos\theta\) for \(0\le\theta\le\pi\). [3]

A tile is a regular hexagon whose corners are the sixth roots of unity, scaled so that the distance from the centre to each corner is 5 cm.

1. Write the corners as complex numbers in exponential form. [1]
2. Find the length of one side, and the area of the tile to 3 significant figures. [2]