A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 9: Trigonometric ratios · Lesson 3

Graphs of sine, cosine and tangent

70 marks

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Quick questions to start.

A1
Last lesson
A triangle has sides $6$ cm and $9$ cm with an angle of $5 0^{\circ}$ between them. Find its area, to 3 s.f.
[2 marks]
Answer
A2
Needed today
Write down the exact values of $cos 3 0^{\circ}$ and $tan 4 5^{\circ}$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$
A3
From chapter 4
The curve $y = f (x)$ has a maximum point at $(2, 5)$ . Write down the maximum point on $y = f (x - 3)$ and on $y = 2 f (x)$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$
A4
From chapter 9
In triangle $A B C$ , $B C = 7$ cm, angle $A = 4 0^{\circ}$ and angle $B = 7 5^{\circ}$ . Find $A C$ , to 3 s.f.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

$cos 5 0^{\circ} = 0.643$ to 3 s.f. Use the graph of $y = cos x$ to find all the values of $x$ in $- 36 0^{\circ} \leq x \leq 36 0^{\circ}$ for which $cos x = 0.643$ .

Hint

y = cos x

is symmetrical about the

y

-axis and repeats every

36 0^{\circ}

B1
$tan 3 5^{\circ} = 0.700$ to 3 s.f. Use the graph of $y = tan x$ to find all the values of $x$ in $- 18 0^{\circ} \leq x \leq 36 0^{\circ}$ for which $tan x = - 0.700$ .
[3 marks]
Answer $x = \dots\dots$

Example 2

For $y = 3 - 2 cos x$ , $0^{\circ} \leq x \leq 36 0^{\circ}$ , find the maximum and minimum values of $y$ and where they occur. State where the curve meets the $y$ -axis.

Hint Because of the minus sign, the largest

y

comes from the smallest

cos x

B2
For $y = 2 - 5 sin x$ , $- 18 0^{\circ} \leq x \leq 18 0^{\circ}$ , find the maximum and minimum values of $y$ and where they occur. State where the curve meets the $y$ -axis.
[3 marks]

Example 3

Sketch $y = tan (x - 6 0^{\circ})$ , $0^{\circ} \leq x \leq 36 0^{\circ}$ . Give the asymptotes and the points where the curve meets the axes.

Hint

y = tan (x - 6 0^{\circ})

y = tan x

moved

6 0^{\circ}

to the right. Move the asymptotes too.

B3
Sketch $y = tan (x + 3 0^{\circ})$ , $0^{\circ} \leq x \leq 36 0^{\circ}$ . Give the asymptotes and the exact points where the curve meets the axes.
[4 marks]

Example 4

Part of $y = sin (x + k)$ , $0^{\circ} < k < 18 0^{\circ}$ , is shown. It meets the $x$ -axis going down at $A (3 0^{\circ}, 0)$ and the $y$ -axis at $C$ . Find $k$ and $C$ exactly.

Not drawn accurately

Hint

sin θ

crosses zero going down when

θ = 18 0^{\circ}

. So

3 0^{\circ} + k = 18 0^{\circ}

B5
$y = cos (x + k)$ , $0^{\circ} < k < 18 0^{\circ}$ , has a minimum point at $P (13 5^{\circ}, - 1)$ . Find $k$ , the exact point where the curve meets the $y$ -axis, and the maximum point for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[4 marks]

Practice

A-Level

C1
Describe fully the single transformation that maps (a) $y = sin x$ onto $y = sin (x - 2 0^{\circ})$ ; (b) $y = cos x$ onto $y = cos \frac{1}{3} x$ ; (c) $y = tan x$ onto $y = tan x - 2$ ; (d) $y = sin x$ onto $y = sin (- x)$ .
[4 marks]

C2
Sketch $y = - 2 cos x$ for $- 18 0^{\circ} \leq x \leq 18 0^{\circ}$ . Give the coordinates of the points where the curve meets the axes, and of its maximum and minimum points.
[4 marks]
C3
$f (x) = 3 sin 2 x + 1$ . Write down the period of $f$ and the range of $f$ . Find how many times the graph of $y = f (x)$ crosses the $x$ -axis for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[3 marks]
Answerperiod $= \dots\dots$ range $\dots\dots$ $\dots\dots$ times

Exam-style questions

Edexcel exam style

D1
The diagram shows the curve $y = tan (x + k)$ , $0^{\circ} \leq x \leq 36 0^{\circ}$ , where $0^{\circ} < k < 9 0^{\circ}$ . The line $x = 3 0^{\circ}$ is an asymptote. The curve meets the $x$ -axis at $A$ and $B$ , and the $y$ -axis at $C$ .
Not drawn accurately
(a)
Find the value of $k$ , and write down the equation of the other asymptote for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[3 marks]
Answer $k = \dots\dots$ $x = \dots\dots$
(b)
Find the coordinates of $A$ , $B$ and $C$ , giving $C$ exactly.
[4 marks]
(c)
Hence solve $tan (x + k) = 1$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[3 marks]
Answer $x = \dots\dots$
Total for D1: 10 marks

D2
Ali writes: "The graph of $y = sin 2 x$ is the graph of $y = sin x$ stretched parallel to the $x$ -axis with scale factor $2$ . So the period of $y = sin 2 x$ is $72 0^{\circ}$ ."
Ali is wrong.
(a)
Explain Ali's mistake and give the correct period. Write down the period of $y = sin \frac{1}{2} x$ .
[3 marks]
(b)
One solution of $sin 2 x = 0.3$ is $x = 8.7 3^{\circ}$ . Using the symmetry and the period of the graph of $y = sin 2 x$ , find the other three solutions for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[3 marks]
Answer $x = \dots\dots$
Total for D2: 6 marks
D4
A big wheel turns at a steady speed. The height, $h$ metres, of a seat above the ground $t$ minutes after it passes the lowest point is modelled by $h = 20 - 18 cos (15 t)^{\circ}$ , $0 \leq t \leq 48$ .
(a)
Write down the greatest height of the seat and the first time it reaches it. Find the time for one complete turn of the wheel.
[3 marks]
Answer $\dots\dots$ m at $t = \dots\dots$ one turn $\dots\dots$ minutes
(b)
Find the times during the first turn when the seat is at least $29$ m above the ground.
[4 marks]
Answer $\dots\dots \leq t \leq \dots\dots$
(c)
Hence find the total time for $0 \leq t \leq 48$ that the seat is at least $29$ m above the ground. Give one reason why the model may not be realistic.
[2 marks]
Total for D4: 9 marks
D5
The diagram shows part of the curve $y = a sin b x + c$ , where $a$ , $b$ and $c$ are positive constants. $P (1 5^{\circ}, 7)$ is a maximum point and $Q (4 5^{\circ}, - 1)$ is the next minimum point.
Not drawn accurately
(a)
Find the values of $a$ , $b$ and $c$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$
(b)
Hence find the number of solutions of $a sin b x + c = 5$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[2 marks]
Answer
Total for D5: 6 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Find all the solutions of $sin 5 x = sin x$ for $0^{\circ} \leq x \leq 18 0^{\circ}$ . Do not use a calculator.
[6 marks]
Hint 1 · What to try
Look at the graph of $y = sin θ$ . Two angles have the same sine when they are equal, or when they add up to $18 0^{\circ}$ , give or take whole turns of $36 0^{\circ}$ .
Hint 2 · The first line
So $5 x = x + 36 0^{\circ} n$ or $5 x = 18 0^{\circ} - x + 36 0^{\circ} n$ , for whole numbers $n$ .
Hint 3 · The full method
The first gives $x = 9 0^{\circ} n$ : $0^{\circ}, 9 0^{\circ}, 18 0^{\circ}$ . The second gives $6 x = 18 0^{\circ} + 36 0^{\circ} n$ , so $x = 3 0^{\circ} + 6 0^{\circ} n$ : $3 0^{\circ}, 9 0^{\circ}, 15 0^{\circ}$ . Together: $0^{\circ}, 3 0^{\circ}, 9 0^{\circ}, 15 0^{\circ}, 18 0^{\circ}$
Answer $x = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can sketch the graphs of sin, cos and tan and mark their key points.
I can use symmetry and period to find more values from one.
I can sketch a stretched or translated trig graph.
I can find the constants in a trig graph from its key points.

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