IB Mathematics AHL 3.10 Compound angle identities AA HL Paper 3- Exam Style Questions

Question

This question investigates the sum of sine and cosine functions.

The expression $ 3 \sin x + 4 \cos x $ can be written in the form $ A \cos(B x + C) + D $, where $ A, B \in \mathbb{R}^+ $, $ C, D \in \mathbb{R} $, and $ -\pi < C \leq \pi $.

The expression $ 5 \sin x + 12 \cos x $ can be written in the form $ A \cos(B x + C) + D $, where $ A, B \in \mathbb{R}^+ $, $ C, D \in \mathbb{R} $, and $ -\pi < C \leq \pi $.

In general, the expression $ a \sin x + b \cos x $ can be written in the form $ A \cos(B x + C) + D $, where $ a, b, A, B \in \mathbb{R}^+ $, $ C, D \in \mathbb{R} $, and $ -\pi < C \leq \pi $.

The expression $ a \sin x + b \cos x $ can also be written as $ \sqrt{a^2 + b^2} \left( \frac{a}{\sqrt{a^2 + b^2}} \sin x + \frac{b}{\sqrt{a^2 + b^2}} \cos x \right) $.

Let $ \frac{a}{\sqrt{a^2 + b^2}} = \sin \theta $.

(a) (i) Sketch the graph of $ y = 3 \sin x + 4 \cos x $, for $ -2 \pi \leq x \leq 2 \pi $. [1]

(ii) Write down the amplitude of this graph. [1]

(iii) Write down the period of this graph. [1]

(b) (i) Use your answers from part (a) to write down the value of $ A $, $ B $, and $ D $. [1]

(ii) Find the value of $ C $. [2]

(ii) Comment on your answer to part (c)(i). [1]

(d) By considering the graph of $ y = 5 \sin x + 12 \cos x $, find the value of $ A $, $ B $, $ C $, and $ D $. [5]

(e) Conjecture an expression, in terms of $ a $ and $ b $, for:

(i) $ A $; [1]

(ii) $ B $; [1]

(iii) $ C $; [1]

(iv) $ D $. [1]

(f) (i) Show that $ \frac{b}{\sqrt{a^2 + b^2}} = \cos \theta $. [2]

(ii) Show that $ \frac{a}{b} = \tan \theta $. [1]

(g) Hence prove your conjectures in part (e). [6]

▶️ Answer/Explanation

Markscheme Solution

(a) (i) Sketch of $ y = 3 \sin x + 4 \cos x $:

$Graph of $ y = 3 \sin x + 4 \cos x $$

Correct graph: (A1)

[1 mark]

(a) (ii) Amplitude:

$ 5 \quad (A1) $

[1 mark]

(a) (iii) Period:

$ 2 \pi \quad (A1) $

[1 mark]

(b) (i) Values of $ A $, $ B $, and $ D $:

$ A = 5, B = 1, D = 0 \quad (A1) $

[1 mark]

(b) (ii) Value of $ C $:

Maximum at $ x = 0.644 $: (M1)

So $ C = -0.644 \quad (A1) $

[2 marks]

(c) (i) Value of $ \arctan \frac{3}{4} $:

$ 0.644 \quad (A1) $

[1 mark]

(c) (ii) Comment:

It appears that $ C = -\arctan \frac{3}{4} $: (A1)

[1 mark]

(d) Values of $ A $, $ B $, $ C $, and $ D $ for $ y = 5 \sin x + 12 \cos x $:

$Graph of $ y = 5 \sin x + 12 \cos x $$

\[ A = 13 \quad (A1) \]

\[ B = 1 \text{ and } D = 0 \quad (A1) \]

Maximum at $ x = 0.395 $: (M1)

\[ C = -0.395 \quad (=-\arctan \frac{5}{12}) \quad (A1) \]

[5 marks]

(e) (i) Conjecture for $ A $:

\[ A = \sqrt{a^2 + b^2} \quad (A1) \]

[1 mark]

(e) (ii) Conjecture for $ B $:

\[ B = 1 \quad (A1) \]

[1 mark]

(e) (iii) Conjecture for $ C $:

\[ C = -\arctan \frac{a}{b} \quad (A1) \]

[1 mark]

(e) (iv) Conjecture for $ D $:

\[ D = 0 \quad (A1) \]

[1 mark]

(f) (i) Show $ \frac{b}{\sqrt{a^2 + b^2}} = \cos \theta $:

Either use a right triangle and Pythagoras’ theorem to show the missing side length is $ b $: (M1)(A1)

Or use $ \sin^2 \theta + \cos^2 \theta = 1 $, leading to the required result: (M1)(A1)

[2 marks]

(f) (ii) Show $ \frac{a}{b} = \tan \theta $:

Either use a right triangle, leading to the required result: (M1)

Or use $ \tan \theta = \frac{\sin \theta}{\cos \theta} $, leading to the required result: (M1)

[1 mark]

(g) Prove conjectures in part (e):

\[ a \sin x + b \cos x = \sqrt{a^2 + b^2} (\sin \theta \sin x + \cos \theta \cos x) \quad (M1) \]

\[ a \sin x + b \cos x = \sqrt{a^2 + b^2} \cos(x – \theta) \quad (M1)(A1) \]

So $ A = \sqrt{a^2 + b^2}, B = 1, D = 0 \quad (A1) $

And $ C = -\theta \quad (M1) $

So $ C = -\arctan \frac{a}{b} \quad (A1) $

[6 marks]

IB Mathematics AHL 3.10 Compound angle identities AA HL Paper 3 | Exam Style Questions

IB Mathematics AHL 3.10 Compound angle identities AA HL Paper 3- Exam Style Questions

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