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

1. 1
   A complex number is $z=\cos\theta+i\sin\theta$, where $\theta$ is real.

   (a)

   Find $z^{-3}$.

   [1 mark]

   • A$\cos3\theta+i\sin3\theta$
   • B$\cos3\theta-i\sin3\theta$
   • C$-\cos3\theta-i\sin3\theta$
   • D$\cos\frac\theta3-i\sin\frac\theta3$

   (b)

   Find $z^4+z^{-4}$.

   [1 mark]

   • A$2i\sin4\theta$
   • B$2\cos^4\theta$
   • C$2\cos4\theta$
   • D$2$

   (c)

   Given that $\theta=\frac\pi9$, find $z^6$ in the form $a+ib$, giving exact values of $a$ and $b$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $c=\cos\theta$ and $s=\sin\theta$, and consider $(c+is)^3$ expanded using the binomial theorem.

   (a)

   Which expression is the real part of $(c+is)^3$?

   [1 mark]

   • A$c^3-3cs^2$
   • B$c^3+3cs^2$
   • C$3c^2s-s^3$
   • D$c^3-3c^2s$

   (b)

   Which expression equals $\sin3\theta$ in terms of $\sin\theta$ only?

   [1 mark]

   • A$3\sin\theta+4\sin^3\theta$
   • B$4\sin\theta-3\sin^3\theta$
   • C$3\sin\theta-4\sin^2\theta$
   • D$3\sin\theta-4\sin^3\theta$

   (c)

   Hence express $\cos3\theta$ in terms of $\cos\theta$ only.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   Show that if the result is true for $n=k$, where $k\ge1$, then it is true for $n=k+1$.

   [3 marks]

   (b)

   Prove that the result is also true for $n=-m$, where $m$ is a positive integer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Throughout, $z=\cos\theta+i\sin\theta$, $c=\cos\theta$ and $s=\sin\theta$.

   (a)

   Show that $\cos5\theta=16\cos^5\theta-20\cos^3\theta+5\cos\theta$.

   [6 marks]

   (b)

   By considering $\left(z-\dfrac1z\right)^4$, show that $\sin^4\theta=\frac18(\cos4\theta-4\cos2\theta+3)$, and hence find $\displaystyle\int_0^{\pi/2}\sin^4\theta\,d\theta$.

   [6 marks]

   Total for question 4: 12 marks

End of questions