This question asks you to investigate some properties of the sequence of functions of…

MATHEMATICS · Subjective · A Level · Mathematics · Trigonometry · Trigonometry

Problem

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

. 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 (b.i) local maximum points; (b.ii) local minimum points; (c) On a new set of axes, sketch the graphs of

y = f_2(x)

and

y = f_4(x)

for

-1 \leq x \leq 1

. 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 (d.i) local maximum points; (d.ii) local minimum points. (e) Solve the equation

f_n'(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

. The sequence of functions,

f_n(x)

, defined above can be expressed as a sequence of polynomials of degree

n

. (f) Use an appropriate trigonometric identity to show that

f_2(x) = 2x^2 - 1

. Consider

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

. (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)

. (h.i) Hence show that

f_{n+1}(x) + f_{n-1}(x) = 2x f_n(x)

n \in \mathbb{Z}^+

. (h.ii) Hence express

f_3(x)

as a cubic polynomial.

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