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QUESTION 16 (6 marks) Consider the identity cos(40) = Acos? (0)+ Bsin? (0)+ C where A Band C € Z a) Determine the values of 4 B and C using De Moivre’s theorem. [5 marks] b) State an appropriate method of verifying your results from 16a). [1 mark]
QUESTION 11 (6 marks) a) Use the binomial theorem to expand (cos(8) + i sin(@))3. Express your result in simplified form. [2 marks] b) Use De Moivre’s theorem to expand (cos(8) + isin(6))3. [1 mark] ¢) Use your results from 11a) and 11b) to prove cos(38) = 4 cos>(0) — 3 cos(8). [3 marks]
QUESTION 13 (5 marks) z| Given z € C where z # 0 prove —— = | zl ZZ
QUESTION 12 (7 marks) Consider the complex number z = -—3 + 2i. a) Determine z using the binomial theorem. Leave your answer in the form a+bi where a be R. b) Convert z into the form of rcis(@) where —z <0 < z. c) Use the result from Question 12b) to determine z using De Moivre’s theorem. Leave your answer in the form of rcis(@) where -a <0 < a. d) Evaluate the reasonableness of your results from Questions 12a) and 12c) noting that the two methods to determine z* should produce the same result. [2 marks] [1 mark] [2 marks] [2 marks]
QUESTION 18 (5 marks) Consider the complex solutions to the following equation where 0 < arg(z) < z. (z+1)(2!4-2 47) —zila 474_73 +2°-z)=1-2 Let w be the solution with the maximum possible real part and w be the solution with the maximum possible imaginary part. 4 Show that Mi EZ. WwW
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