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

Exam questions on De Moivre's theorem and trigonometric identities

1. 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
2. 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
3. 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