Specialist Maths — Question 16

QCAA 2020, Paper 2 · 6 marks

Q16 · 2020 · Technology-activeComplex familiar6 marks

QUESTION 16 (6 marks)

Consider the identity

\cos(4\theta)=A\cos^4(\theta)+B\sin^2(\theta)+C,

where

A,B,C\in\mathbb Z

Determine the values of A, B and C using De Moivre’s theorem.

[5 marks]

State an appropriate method of verifying your results from 16a).

[1 mark]

WORKED SOLUTION

6 marks

ANSWER

A=8

B=8

C=-7

Worked solution

a) By De Moivre,

(\cos\theta+i\sin\theta)^4=\cos4\theta+i\sin4\theta

. Equating real parts and using

\sin^2\theta=1-\cos^2\theta

gives

\cos4\theta=8\cos^4\theta+8\sin^2\theta-7,

A=8

B=8

C=-7

b) Graph

y=\cos4\theta

and

y=8\cos^4\theta+8\sin^2\theta-7

and confirm that the graphs coincide.

correctly uses De Moivre’s theorem

[1 mark]

uses binomial expansion with the real parts and simplifies the expression

[1 mark]

establishes a simplified expression following the use of a suitable Pythagorean identity

[1 mark]

establishes a simplified expression in the form

A\cos^4(\theta)+B\sin^2(\theta)+C

[1 mark]

communicates the values of A, B and C

[1 mark]

describes an appropriate verification strategy

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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Compare your working with the guide above.