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Date	May Example questions	Marks available	1	Reference code	EXM.3.AHL.TZ0.5
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Show that	Question number	5	Adapted from	N/A

Question

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]

Markscheme

[1 mark]

a.i.

5 A1

[1 mark]

a.ii.

$2\pi$ A1

[1 mark]

a.iii.

$A = 5$ , $B = 1$ , $D = 0$ A1

[1 mark]

b.i.

maximum at $x = 0.644$ M1

So $C = -0.644$ A1

[2 marks]

b.ii.

0.644 A1

[1 mark]

c.i.

it appears that $C = - {\text{arctan}}\frac{3}{4}$ A1

[1 mark]

c.ii.

$A = 13$ A1

$B = 1$ and $D = 0$ A1

maximum at $x = 0.395$ M1

So C = −0.395 $\left( { = - {\text{arctan}}\frac{5}{{12}}} \right)$ A1

[5 marks]

$A = \sqrt {{a^2} + {b^2}}$ A1

[1 mark]

e.i.

$B = 1$ A1

[1 mark]

e.ii.

$C = - {\text{arctan}}\frac{a}{b}$ A1

[1 mark]

e.iii.

$D = 0$ A1

[1 mark]

e.iv.

EITHER

use of a right triangle and Pythgoras’ to show the missing side length is $b$ M1A1

Use of ${\text{si}}{{\text{n}}^2}\theta + {\text{co}}{{\text{s}}^2}\theta = 1$ , leading to the required result M1A1

[2 marks]

f.i.

EITHER

use of a right triangle, leading to the required result. M1

Use of ${\text{tan}}\,\theta = \frac{{{\text{sin}}\,\theta }}{{{\text{cos}}\,\theta }}$ , leading to the required result. M1

[1 mark]

f.ii.

$a\,{\text{sin}}\,x + b\,{\text{cos}}\,x = \sqrt {{a^2} + {b^2}} \left( {{\text{sin}}\,\theta \,{\text{sin}}\,x + {\text{cos}}\,\theta \,{\text{cos}}\,x} \right)$ M1

$a\,{\text{sin}}\,x + b\,{\text{cos}}\,x = \sqrt {{a^2} + {b^2}} \left( {{\text{cos}}\left( {x - \theta } \right)} \right)$ M1A1

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

And $C = - \theta$ M1

So $C = - {\text{arctan}}\frac{a}{b}$ A1

[6 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

a.iii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

e.i.

[N/A]

e.ii.

[N/A]

e.iii.

[N/A]

e.iv.

[N/A]

f.i.

[N/A]

f.ii.

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations

Show 89 related questions

19M.2.SL.TZ1.S_8c.i:
Find the value of $b$ and of $c$ .
SPM.2.SL.TZ0.9a:
Find the period of $f$ .
22M.2.SL.TZ2.8b:
Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .
EXM.3.AHL.TZ0.5a.iii:
Write down the period of this graph
18M.2.SL.TZ1.S_10a:
Find the coordinates of A.
18M.2.SL.TZ1.S_10b.i:
For the graph of $f$ , write down the amplitude.
19M.2.AHL.TZ1.H_10a:
Write down the maximum and minimum value of $v$ .
17M.2.SL.TZ2.S_4c:
Use the model to find the depth of the water 10 hours after high tide.
19M.2.SL.TZ1.S_8b:
Find the range of $g$ .
17N.2.SL.TZ0.S_10a:
Show that $f(2\pi ) = 2\pi$ .
19M.2.AHL.TZ1.H_10c:
Sketch the graph of $y = p\left( t \right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.
17M.2.SL.TZ1.S_8a.i:
How much time is there between the first low tide and the next high tide?
EXM.3.AHL.TZ0.5c.ii:
Comment on your answer to part (c)(i).
EXM.3.AHL.TZ0.5e.i:
$A$ .
EXM.3.AHL.TZ0.5b.ii:
Find the value of $C$ .
17N.2.SL.TZ0.S_10d:
A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.
EXM.3.AHL.TZ0.5e.iii:
$C$ .
18M.2.SL.TZ1.S_10e:
Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².
19M.1.AHL.TZ2.H_9a:
Find the $x$ -coordinates of the points of intersection of the two graphs.
19M.2.AHL.TZ1.H_10d:
Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left( t \right)$ ≥ 3.
17M.2.SL.TZ1.S_8a.ii:
Find the difference in height between low tide and high tide.
17N.2.SL.TZ0.S_10b.ii:
Find the equation of $L$ .
18M.2.SL.TZ1.S_10d:
Find the maximum speed of the ball.
19M.2.AHL.TZ1.H_10g:
Given that $p\left( t \right)$ can be written as $p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d$ where $a$ , $b$ , $c$ , $d$ > 0, use your graph to find the values of $a$ , $b$ , $c$ and $d$ .
EXM.3.AHL.TZ0.5a.i:
Sketch the graph $y = 3\,{\text{sin}}\,x + 4\,{\text{cos}}\,x$ , for $- 2\pi \leqslant x \leqslant 2\pi$
16N.2.SL.TZ0.S_10a:
(i) Find the value of $c$ .

(ii) Show that $b = \frac{\pi }{6}$ .

(iii) Find the value of $a$ .
19M.1.SL.TZ2.S_7b:
Consider the graph of $g\left( x \right) = 2\,{\text{sin}}\,px$ , 0 ≤ $x$ < $2\pi$ , where $p$ > 0.

Find the greatest value of $p$ such that the graph of $g$ does not intersect the line $y = - 1$ .
EXM.3.AHL.TZ0.5c.i:
Find ${\text{arctan}}\frac{3}{4}$ , giving the answer to 3 significant figures.
EXM.3.AHL.TZ0.5g:
Hence prove your conjectures in part (e).
17N.2.SL.TZ0.S_10b.i:
Find the coordinates of ${{\text{P}}_0}$ and of ${{\text{P}}_1}$ .
22M.1.SL.TZ1.6b:
The $y$ -intercept of the graph of $g$ is at $(0, r)$ .

Given that $g (x) \geq 7$ , find the smallest value of $r$ .
EXN.1.SL.TZ0.9b:
Find the minimum height of the ball above the ground.
EXN.1.SL.TZ0.9d:
For the first 2 seconds of its motion, determine the amount of time that the ball is less than $1.8 + 0.2 \sqrt{2}$ metres above the ground.
17N.2.SL.TZ0.S_10c:
Show that the distance between the $x$ -coordinates of ${{\text{P}}_k}$ and ${{\text{P}}_{k + 1}}$ is $2\pi$ .
19M.2.AHL.TZ1.H_10b:
Write down two transformations that will transform the graph of $y = v\left( t \right)$ onto the graph of $y = i\left( t \right)$ .
22M.2.AHL.TZ2.10b:
Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .
20N.2.SL.TZ0.S_8a.iii:
Hence find the number of rotations the water wheel makes in one hour.
EXM.3.AHL.TZ0.5b.i:
Use your answers from part (a) to write down the value of $A$ , $B$ and $D$ .
20N.2.SL.TZ0.S_8b:
Find $k$ .
20N.2.SL.TZ0.S_8d:
Determine the rate of change of $h$ when the top of the bucket is at $B$ .
EXM.3.AHL.TZ0.5f.i:
Show that $\frac{b}{{\sqrt {{a^2} + {b^2}} }} = {\text{cos}}\,\theta$ .
19M.2.SL.TZ1.S_8c.ii:
Find the period of $g$ .
19M.2.SL.TZ1.S_8d:
The equation $g\left( x \right) = 12$ has two solutions where $\pi$ ≤ $x$ ≤ ${\frac{{4\pi }}{3}}$ . Find both solutions.
18M.2.SL.TZ1.S_10b.ii:
For the graph of $f$ , write down the period.
19M.1.AHL.TZ2.H_9c:
At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$ -coordinate of A on the graph of $g$ .
17M.2.SL.TZ2.S_4a:
Find the value of $p$ .
SPM.2.SL.TZ0.9b.ii:
Hence, find the value of $f$ (6).
21M.1.SL.TZ2.8a:
Find the $x$ -coordinates of $A$ and $B$ .
19M.1.AHL.TZ2.H_9b:
Find the exact area of the shaded region, giving your answer in the form $p\pi + q\sqrt 3$ , where $p$ , $q \in \mathbb{Q}$ .
17M.2.SL.TZ1.S_8b.ii:
Find the value of $q$ ;
SPM.2.SL.TZ0.9b.i:
Find the value of $b$ .
EXN.1.SL.TZ0.9a:
Find the height of the ball above the ground when it is released.
EXN.1.SL.TZ0.9c:
Show that the ball takes $2$ seconds to return to its initial height above the ground for the first time.
20N.2.SL.TZ0.S_8c:
Find $A \hat{O} B$ .
19M.2.AHL.TZ1.H_10e:
Find ${p_{av}}$ (0.007).
20N.2.SL.TZ0.S_8a.ii:
Calculate the number of seconds it takes for the water wheel to complete one rotation.
20N.2.SL.TZ0.S_8a.i:
Find the height of point $A$ above the ground.
19M.2.AHL.TZ1.H_10f:
With reference to your graph of $y = p\left( t \right)$ explain why ${p_{av}}\left( T \right)$ > 0 for all $T$ > 0.
17M.2.SL.TZ2.S_4b:
Find the value of $q$ .
19M.2.SL.TZ1.S_8a:
The range of $f$ is $k$ ≤ $f\left( x \right)$ ≤ $m$ . Find $k$ and $m$ .
21M.2.SL.TZ2.7d:
Find the value of $p$ and the value of $q$ .
16N.2.SL.TZ0.S_10c:
(i) Find $w$ .

(ii) Hence or otherwise, find the maximum positive rate of change of $g$ .
21M.2.SL.TZ2.7e:
If the observer stands directly under point $C$ for one minute, point $D$ will pass over his head $n$ times.

Find the value of $n$ .
21N.2.SL.TZ0.8a:
Show that $b = \frac{π}{6}$ .
21N.2.SL.TZ0.8c:
Find the value of $d$ .
21N.2.SL.TZ0.8d:
Find the smallest possible value of $c$ .
22M.1.SL.TZ1.6a:
Describe these two transformations.
21N.2.SL.TZ0.8b:
Find the value of $a$ .
21N.2.AHL.TZ0.9a:
Show that $b = \frac{π}{6}$ .
21N.2.AHL.TZ0.9b:
Find the value of $a$ .
21N.2.AHL.TZ0.9d:
Find the smallest possible value of $c$ .
EXM.3.AHL.TZ0.5a.ii:
Write down the amplitude of this graph
EXM.3.AHL.TZ0.5d:
By considering the graph of $y = 5\,{\text{sin}}\,x + 12\,{\text{cos}}\,x$ , find the value of $A$ , $B$ , $C$ and $D$ .
18M.2.SL.TZ1.S_10c:
Hence, write $f\left( x \right)$ in the form $p\,\,{\text{cos}}\,\left( {x + r} \right)$ .
17M.2.SL.TZ1.S_8b.i:
Find the value of $p$ ;
EXM.3.AHL.TZ0.5e.ii:
$B$ .
EXM.3.AHL.TZ0.5e.iv:
$D$ .
17M.2.SL.TZ1.S_8b.iii:
Find the value of $r$ .
17M.2.SL.TZ1.S_8c:
There are two high tides on 12 December 2017. At what time does the second high tide occur?
16N.2.SL.TZ0.S_10b:
(i) Write down the value of $k$ .

(ii) Find $g(x)$ .
SPM.2.SL.TZ0.9d:
Find the value of $x$ for which the functions have the greatest difference.
SPM.2.SL.TZ0.9c:
Find the value of $p$ and the value of $q$ .
21M.2.SL.TZ1.4:
A Ferris wheel with diameter $110$ metres rotates at a constant speed. The lowest point on the wheel is $10$ metres above the ground, as shown on the following diagram. $P$ is a point on the wheel. The wheel starts moving with $P$ at the lowest point and completes one revolution in $20$ minutes.

The height, $h$ metres, of $P$ above the ground after $t$ minutes is given by $h (t) = a \cos (b t) + c$ , where $a, b, c \in ℝ$ .

Find the values of $a$ , $b$ and $c$ .
21N.2.AHL.TZ0.9e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9c:
Find the value of $d$ .
21N.2.AHL.TZ0.9f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9g:
A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.

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Markscheme

Examiners report

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