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HL Paper 3

This question investigates the sum of sine and cosine functions

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

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

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

Conjecture an expression, in terms of $a$ and $b$ , for

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

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

Sketch the graph $y = 3\,{\text{sin}}\,x + 4\,{\text{cos}}\,x$ , for $- 2\pi \leqslant x \leqslant 2\pi$

[1]

a.i.

Write down the amplitude of this graph

[1]

a.ii.

Write down the period of this graph

[1]

a.iii.

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

[1]

b.i.

Find the value of $C$ .

[2]

b.ii.

Find ${\text{arctan}}\frac{3}{4}$ , giving the answer to 3 significant figures.

[1]

c.i.

Comment on your answer to part (c)(i).

[1]

c.ii.

By considering the graph of $y = 5\,{\text{sin}}\,x + 12\,{\text{cos}}\,x$ , find the value of $A$ , $B$ , $C$ and $D$ .

[5]

$A$ .

[1]

e.i.

$B$ .

[1]

e.ii.

$C$ .

[1]

e.iii.

$D$ .

[1]

e.iv.

Show that $\frac{b}{{\sqrt {{a^2} + {b^2}} }} = {\text{cos}}\,\theta$ .

[2]

f.i.

Show that $\frac{a}{b} = {\text{tan}}\,\theta$ .

[1]

f.ii.

Hence prove your conjectures in part (e).

[6]

This question asks you to investigate some properties of the sequence of functions of the form ${f_n}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right)$ , −1 ≤ $x$ ≤ 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.

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

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

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

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

On the same set of axes, sketch the graphs of $y = {f_1}(x)$ and $y = {f_3}(x)$ for −1 ≤ $x$ ≤ 1.

[2]

local maximum points;

[3]

b.i.

local minimum points;

[1]

b.ii.

On a new set of axes, sketch the graphs of $y = {f_2}(x)$ and $y = {f_4}(x)$ for −1 ≤ $x$ ≤ 1.

[2]

local maximum points;

[3]

d.i.

local minimum points.

[1]

d.ii.

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 $x = {\text{cos}}\frac{{k\pi }}{n}$ where $k \in {\mathbb{Z}^ + }$ and 0 < $k$ < $n$ .

[4]

Use an appropriate trigonometric identity to show that ${f_2}(x) = 2{x^2} - 1$ .

[2]

Use an appropriate trigonometric identity to show that ${f_{n + 1}}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) - {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)$ .

[2]

Hence show that ${f_{n + 1}}(x) + {f_{n - 1}}(x) = 2x{f_n}\left( x \right)$ , $n \in {\mathbb{Z}^ + }$ .

[3]

h.i.

Hence express ${f_3}(x)$ as a cubic polynomial.

[2]

h.ii.

This question asks you to examine various polygons for which the numerical value of the area is the same as the numerical value of the perimeter. For example, a $3$ by $6$ rectangle has an area of $18$ and a perimeter of $18$ .

For each polygon in this question, let the numerical value of its area be $A$ and let the numerical value of its perimeter be $P$ .

An $n$ -sided regular polygon can be divided into $n$ congruent isosceles triangles. Let $x$ be the length of each of the two equal sides of one such isosceles triangle and let $y$ be the length of the third side. The included angle between the two equal sides has magnitude $\frac{2 π}{n}$ .

Part of such an $n$ -sided regular polygon is shown in the following diagram.

Consider a $n$ -sided regular polygon such that $A = P$ .

The Maclaurin series for $\tan x$ is $x + \frac{x^{3}}{3} + \frac{2 x^{5}}{15} + \dots$

Consider a right-angled triangle with side lengths $a, b$ and $\sqrt{a^{2} + b^{2}}$ , where $a \geq b$ , such that $A = P$ .

Find the side length, $s$ , where $s > 0$ , of a square such that $A = P$ .

[3]

Write down, in terms of $x$ and $n$ , an expression for the area, $A_{T}$ , of one of these isosceles triangles.

[1]

Show that $y = 2 x \sin \frac{π}{n}$ .

[2]

Use the results from parts (b) and (c) to show that $A = P = 4 n \tan \frac{π}{n}$ .

[7]

Use the Maclaurin series for $\tan x$ to find $\lim_{n \to \infty} (4 n \tan \frac{π}{n})$ .

[3]

e.i.

Interpret your answer to part (e)(i) geometrically.

[1]

e.ii.

Show that $a = \frac{8}{b - 4} + 4$ .

[7]

By using the result of part (f) or otherwise, determine the three side lengths of the only two right-angled triangles for which $a, b, A, P \in ℤ$ .

[3]

g.i.

Determine the area and perimeter of these two right-angled triangles.

[1]

g.ii.