Analysis and Approaches for IBDP Maths Ebook 2 | Page 127

39
Paper 2 Section B – De Moivre’s Theorem
Example
(a) (i) Use the binomial theorem to expand
(cos�
isin )
6
� � .
(ii) Hence, express cos6� and sin 6� in terms of sin� and cos� .
Let z�r(cos�� isin �)
, where � is measured in radians, be the solution of
which has the smallest positive argument.
6
z � �
i 0
[6]
(b) Find the values of r and � .
(c) Hence, show that
(d) Show that
Solution
� � �
� � � � .
12 12 12
6 4 2
tan 15 tan 15 tan 1 0
� � � �� � � �
� �� �
� 12 12 �� 12 12 �
2 2 2 2
3cos �sin cos �3sin � 2
.
[4]
[3]
[4]
9
(a)
(i)
(cos�
� isin �)
5 6
� � � �
6
6 �6�
5
�cos � � � � i cos � sin�
�1�
� 6�
2 4 2 �6�
3 3 3
�� �i cos � sin � �� �i cos � sin �
� 2�
� 3�
� 6�
4 2 4 �6�
5 5
�� �i cos � sin � �� �i cos� sin �
� 4�
�5�
6 6
�i sin �
6 5 4 2
� cos � � 6i cos � sin� �15cos � sin �
3 3 2 4
�20i cos � sin � �15cos � sin �
�6i cos sin sin
A1
A1
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