IB Mathematics AHL 3.9 Reciprocal trigonometric ratios AA HL Paper 3 -Exam Style Questions

Question

This question asks you to investigate some properties of the sequence of functions of the form $ f_n(x) = \cos(n \arccos x) $, $ -1 \leq x \leq 1 $, and $ n \in \mathbb{Z}^+ $.

Important: When sketching graphs in this question, you are not required to find the coordinates of any axes intercepts or the coordinates of any stationary points unless requested.

(a) On the same set of axes, sketch the graphs of $ y = f_1(x) $ and $ y = f_3(x) $ for $ -1 \leq x \leq 1 $. [2]

(b) For odd values of $ n > 2 $, use your graphic display calculator to systematically vary the value of $ n $. Hence suggest an expression for odd values of $ n $ describing, in terms of $ n $, the number of:

(i) local maximum points; [3]

(ii) local minimum points. [1]

(d) For even values of $ n > 2 $, use your graphic display calculator to systematically vary the value of $ n $. Hence suggest an expression for even values of $ n $ describing, in terms of $ n $, the number of:

(i) local maximum points; [3]

(ii) local minimum points. [1]

The sequence of functions, $ f_n(x) $, defined above can be expressed as a sequence of polynomials of degree $ n $.

Consider $ f_{n+1}(x) = \cos((n+1) \arccos x) $.

(e) Solve the equation $ f_n^{\prime}(x) = 0 $ and hence show that the stationary points on the graph of $ y = f_n(x) $ occur at $ x = \cos \frac{k \pi}{n} $ where $ k \in \mathbb{Z}^+ $ and $ 0 < k < n $. [4]

(f) Use an appropriate trigonometric identity to show that $ f_2(x) = 2 x^2 – 1 $. [2]

(g) Use an appropriate trigonometric identity to show that $ f_{n+1}(x) = \cos(n \arccos x) \cos(\arccos x) – \sin(n \arccos x) \sin(\arccos x) $. [2]

(h) (i) Hence show that $ f_{n+1}(x) + f_{n-1}(x) = 2 x f_n(x) $, $ n \in \mathbb{Z}^+ $. [3]

(ii) Hence express $ f_3(x) $ as a cubic polynomial. [2]

▶️ Answer/Explanation

Markscheme Solution

(a) Sketch of $ y = f_1(x) $ and $ y = f_3(x) $:

$Graphs of $ y = f_1(x) $ and $ y = f_3(x) $$

Correct graph of $ y = f_1(x) $: (A1)

Correct graph of $ y = f_3(x) $: (A1)

[2 marks]

(b) (i) Local maximum points for odd $ n > 2 $:

Graphical or tabular evidence that $ n $ has been systematically varied: (M1)

e.g., $ n = 3 $: 1 local maximum point and 1 local minimum point;

$ n = 5 $: 2 local maximum points and 2 local minimum points;

$ n = 7 $: 3 local maximum points and 3 local minimum points: (A1)

Number of local maximum points: $ \frac{n-1}{2} $ (A1)

[3 marks]

(b) (ii) Local minimum points for odd $ n > 2 $:

Number of local minimum points: $ \frac{n-1}{2} $ (A1)

Note: Allow follow through from an incorrect local maximum formula expression.

[1 mark]

(c) Sketch of $ y = f_2(x) $ and $ y = f_4(x) $:

$Graphs of $ y = f_2(x) $ and $ y = f_4(x) $$

Correct graph of $ y = f_2(x) $: (A1)

Correct graph of $ y = f_4(x) $: (A1)

[2 marks]

(d) (i) Local maximum points for even $ n > 2 $:

Graphical or tabular evidence that $ n $ has been systematically varied: (M1)

e.g., $ n = 2 $: 0 local maximum points and 1 local minimum point;

$ n = 4 $: 1 local maximum point and 2 local minimum points;

$ n = 6 $: 2 local maximum points and 3 local minimum points: (A1)

Number of local maximum points: $ \frac{n-2}{2} $ (A1)

[3 marks]

(d) (ii) Local minimum points for even $ n > 2 $:

Number of local minimum points: $ \frac{n}{2} $ (A1)

[1 mark]

(e) Stationary points of $ y = f_n(x) $:

\[ f_n(x) = \cos(n \arccos x) \]

\[ f_n^{\prime}(x) = \frac{n \sin(n \arccos x)}{\sqrt{1-x^2}} \quad (M1)(A1) \]

Note: Award (M1) for attempting to use the chain rule.

\[ f_n^{\prime}(x) = 0 \implies n \sin(n \arccos x) = 0 \quad (M1) \]

\[ n \arccos x = k \pi \quad (k \in \mathbb{Z}^+) \quad (A1) \]

Leading to:

\[ x = \cos \frac{k \pi}{n} \quad (k \in \mathbb{Z}^+, 0 < k < n) \]

[4 marks]

(f) Show $ f_2(x) = 2 x^2 – 1 $:

\[ f_2(x) = \cos(2 \arccos x) \]

\[ = 2 (\cos(\arccos x))^2 – 1 \quad (M1) \]

Stating that $ \cos(\arccos x) = x $: (A1)

So $ f_2(x) = 2 x^2 – 1 $

[2 marks]

(g) Show $ f_{n+1}(x) = \cos(n \arccos x) \cos(\arccos x) – \sin(n \arccos x) \sin(\arccos x) $:

\[ f_{n+1}(x) = \cos((n+1) \arccos x) \]

\[ = \cos(n \arccos x + \arccos x) \quad (A1) \]

Use of $ \cos(A + B) = \cos A \cos B – \sin A \sin B $: (M1)

\[ = \cos(n \arccos x) \cos(\arccos x) – \sin(n \arccos x) \sin(\arccos x) \]

[2 marks]

(h) (i) Show $ f_{n+1}(x) + f_{n-1}(x) = 2 x f_n(x) $:

\[ f_{n-1}(x) = \cos((n-1) \arccos x) \quad (A1) \]

\[ = \cos(n \arccos x) \cos(\arccos x) + \sin(n \arccos x) \sin(\arccos x) \quad (M1) \]

\[ f_{n+1}(x) + f_{n-1}(x) = 2 \cos(n \arccos x) \cos(\arccos x) \quad (A1) \]

\[ = 2 x f_n(x) \]

[3 marks]

(h) (ii) Express $ f_3(x) $ as a cubic polynomial:

\[ f_3(x) = 2 x f_2(x) – f_1(x) \quad (M1) \]

\[ = 2 x (2 x^2 – 1) – x \]

\[ = 4 x^3 – 3 x \quad (A1) \]

[2 marks]

IB Mathematics AHL 3.9 Reciprocal trigonometric ratios AA HL Paper 3 | Exam Style Questions

IB Mathematics AHL 3.9 Reciprocal trigonometric ratios AA HL Paper 3 -Exam Style Questions

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