Analysis and Approaches for IBDP Maths Ebook 2 | Page 130

Your Practice Set – Analysis and Approaches for IBDP Mathematics
2. (a) Use De Moivre’s theorem to express sin8� in terms of sin� and cos� .
Let zi �r(cos�i � isin �i)
, where �
i
is measured in radians and i � 1, 2, 3 and
8
Re( zi
) � 0, be the solutions of z �6561 � 0 which has the largest, the second largest and
the third largest negative arguments respectively.
(b) Find the values of r and �
1
.
(c)
2 4 6
Hence, show that 1�7 tan � � 7 tan � � tan � � 0 .
1 1 1
Let A , B , C , D , E and F be the points on an Argand diagram representing the
complex numbers z
1
, z
2
, z
3
, z � 3
, z � 2
and z � 1
respectively. Let M be the circle centred
at the origin O and passes through the above six points.
(d) (i) Write down the radius of M .
(ii) Write down the shape of the quadrilateral BCDE .
[5]
[4]
[3]
(iii)
Write down the shape of the quadrilateral ACDF.
(iv) Find the length of BE .
[7]
2n
( �cos � isin ) � cos 2n
� isin 2
3. (a) Use mathematical induction to prove that � � � n�
�
for n� .
[6]
(b) Hence, express cos8� in terms of sin� and cos� .
[5]
(c) Hence, or otherwise, solve the equation
4 4 2 6 8
�
cos8� � 41cos � sin � �28cos � sin � � sin � for 0 ��
� .
2
[6]
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