Standard +0.8 This is a Further Maths question requiring students to multiply complex numbers in polar form, apply De Moivre's theorem to find the argument of the product (5π/12), then extract tan(5π/12) by converting to Cartesian form and manipulating surds. While the individual steps are standard, the multi-stage process (multiply → simplify trigonometric expressions → rationalize/simplify surds → extract tangent) and the need to work fluently with exact values makes this moderately challenging, though still within expected Further Maths territory.
The complex numbers \(z\) and \(w\) are defined by
$$z = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4}$$
and
$$w = \cos\frac{\pi}{6} + i\sin\frac{\pi}{6}$$
By evaluating the product \(zw\), show that
$$\tan\frac{5\pi}{12} = 2 + \sqrt{3}$$
[6 marks]
Question 10:
10 | Forms the product zw | 1.1a | M1 |
2 2 3 1
z w = + i + i
2 2 2 2
6 2 6 2
= – + i +
4 4 4 4
Also
π π π π
z w = c o s + + i s in +
4 6 4 6
5 π 5 π
= c o s + i s in
1 2 1 2
π 5 6 2
s in +
5 π
1 2π 4 4
t a n = =
1 2 5 6 2
c o s –
1 2 4 4
= 2 + 3
Obtains
6 2 6 2
– + i +
4 4 4 4 | 1.1b | A1
States or uses
a r g ( z w ) = a r g ( z ) + a r g ( w ) | 3.1a | M1
π π 5 π
States + =
4 6 1 2 | 1.1b | B1
Uses their
6 2 6 2
z w = – + i +
4 4 4 4
to deduce an expression for
5π
tan
12 | 2.2a | M1
Completes a reasoned
argument to obtain
5 π
t a n = 2 + 3
1 2
AG | 2.1 | R1
Question total | 6
Q | Marking instructions | AO | Marks | Typical solution
The complex numbers $z$ and $w$ are defined by
$$z = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4}$$
and
$$w = \cos\frac{\pi}{6} + i\sin\frac{\pi}{6}$$
By evaluating the product $zw$, show that
$$\tan\frac{5\pi}{12} = 2 + \sqrt{3}$$
[6 marks]
\hfill \mbox{\textit{AQA Further Paper 1 2024 Q10 [6]}}