AQA Level 2 Further Maths · Level 2 Certificate · Chapter 6: Geometry I · Lesson 5

Trigonometric functions and their graphs

About 75 minutesNo calculator63 marks

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Date

Do Now

Four questions to start. Two of them use what you need today.

A1
Needed today
Work out the exact value of $sin 6 0^{\circ} \times tan 3 0^{\circ} + cos 4 5^{\circ} \times sin 4 5^{\circ}$ .
[2 marks]
Answer
A2
Last lesson
A ladder $8$ m long leans against a vertical wall. It makes an angle of $6 0^{\circ}$ with the horizontal ground. Work out exactly how high up the wall the ladder reaches.
[2 marks]
Answer
A3
From chapter 5
A circle has equation $x^{2} + y^{2} - 6 x + 4 y = 12$ . Work out the centre and the radius of the circle.
[2 marks]
Answercentre $(\dots, \dots)$ radius $\dots$
A4
From chapter 4
$(x - 2)$ is a factor of $x^{3} + k x - 6$ . Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$

Example, then your turn

Level 2

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

Example 1

Work out the exact value of $cos 3 0^{\circ} + cos 15 0^{\circ} + sin 24 0^{\circ}$ . Give your answer in its simplest form.

Hint Write each ratio as

\pm

a ratio of an acute angle:

cos 15 0^{\circ} = - cos 3 0^{\circ}

and

sin 24 0^{\circ} = - sin 6 0^{\circ}

B1
Work out the exact value of $sin 4 5^{\circ} \times cos 13 5^{\circ} + tan 24 0^{\circ}$ . Give your answer in its simplest form.
[3 marks]
Answer

Example 2

The point $P$ lies on the circle $x^{2} + y^{2} = 1$ , and $O P$ makes an angle of $12 0^{\circ}$ with the positive $x$ -axis. Write down the exact coordinates of $P$ . Hence write down the exact values of $sin 12 0^{\circ}$ , $cos 12 0^{\circ}$ and $tan 12 0^{\circ}$ .

Hint

O P

makes

6 0^{\circ}

with the negative

x

-axis.

P

(cos 12 0^{\circ}, sin 12 0^{\circ})

, and

tan 12 0^{\circ} = \frac{y}{x}

B2
The point $P$ lies on the circle $x^{2} + y^{2} = 1$ , and $O P$ makes an angle of $22 5^{\circ}$ with the positive $x$ -axis. Write down the exact coordinates of $P$ . Hence write down the exact values of $sin 22 5^{\circ}$ , $cos 22 5^{\circ}$ and $tan 22 5^{\circ}$ .
[3 marks]
Answer

Example 3

Sketch the graph of $y = cos x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ on the axes. Write down the coordinates of the points where it meets the axes, and of its minimum point.

Hint Start at

(0, 1)

. The cosine graph is the sine graph moved

9 0^{\circ}

to the left.

B3
Sketch the graph of $y = tan x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ on the axes. Write down the equations of its asymptotes, and the coordinates of the points where it crosses the $x$ -axis.
[3 marks]
Answer

Example 4

Here is the graph of $y = sin x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . You are given that $sin 3 5^{\circ} = p$ . Work out the two values of $x$ , for $0^{\circ} \leq x \leq 36 0^{\circ}$ , for which $sin x = - p$ .

Hint The curve between

18 0^{\circ}

and

36 0^{\circ}

is the curve between

0^{\circ}

and

18 0^{\circ}

turned upside down.

B4
Here is the graph of $y = cos x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . You are given that $cos 7 0^{\circ} = q$ . Work out the two values of $x$ , for $0^{\circ} \leq x \leq 36 0^{\circ}$ , for which $cos x = - q$ .
[2 marks]
Answer $x = \dots\dots$ and $x = \dots\dots$

Example 5

Work out the exact values of $sin 48 0^{\circ}$ , $cos (- 12 0^{\circ})$ and $tan 57 0^{\circ}$ .

Hint Add or take away

36 0^{\circ}

until the angle is between

0^{\circ}

and

36 0^{\circ}

; the tangent also repeats every

18 0^{\circ}

B5
Work out the exact values of $cos 39 0^{\circ}$ , $sin (- 15 0^{\circ})$ and $tan (- 22 5^{\circ})$ .
[3 marks]
Answer

Practice

Level 2

C1
Write true or false for each statement. Give a reason for each one.
(i) $sin 5 0^{\circ} = sin 13 0^{\circ}$
(ii) $cos 4 0^{\circ} = cos 14 0^{\circ}$
(iii) $tan 2 0^{\circ} = tan 16 0^{\circ}$
(iv) $sin 25 0^{\circ} = sin 29 0^{\circ}$
[4 marks]
Answer(i) $\dots$ (ii) $\dots$ (iii) $\dots$ (iv) $\dots$
C2
The angle $θ$ is between $18 0^{\circ}$ and $27 0^{\circ}$ . For each of $sin θ$ , $cos θ$ and $tan θ$ , write down whether it is positive or negative.
[2 marks]
Answer

C3
Here is the graph of $y = sin x$ for $- 36 0^{\circ} \leq x \leq 36 0^{\circ}$ . Use it to find how many solutions each equation has for $- 36 0^{\circ} \leq x \leq 36 0^{\circ}$ .
(i) $sin x = - 0.7$ (ii) $sin x = 0$
[2 marks]
Answer(i) $\dots$ (ii) $\dots$
C4
The point $P (- 0.6, - 0.8)$ lies on the circle $x^{2} + y^{2} = 1$ . $O P$ makes the angle $θ$ with the positive $x$ -axis. Write down the values of $sin θ$ , $cos θ$ and $tan θ$ .
[2 marks]
Answer
C5
Explain why $tan 9 0^{\circ}$ has no value. Write down the next angle after $9 0^{\circ}$ for which $tan x$ has no value.
[2 marks]
Answernext angle $\dots\dots$
C6
Describe fully the single transformation that maps the graph of $y = sin x$ onto the graph of $y = cos x$ .
[2 marks]
Answer

Exam-style questions

AQA exam style

D1
Ravi says, " $sin 20 0^{\circ} = sin 2 0^{\circ}$ , because the sine graph repeats itself every $18 0^{\circ}$ ."
Explain why Ravi is wrong. Write $sin 20 0^{\circ}$ in terms of $sin 2 0^{\circ}$ .
[2 marks]
Answer $sin 20 0^{\circ} = \dots\dots$

D2
Here is a sketch of $y = sin x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . $A$ is the maximum point, $C$ is the minimum point and $B$ is where the curve crosses the $x$ -axis between them. The point $P (30, \frac{1}{2})$ lies on the curve.
(a)
Write down the coordinates of $A$ , $B$ and $C$ .
[2 marks]
Answer $A (\dots, \dots)$ $B (\dots, \dots)$ $C (\dots, \dots)$
(b)
The points $Q$ and $R$ also lie on the curve. They both have $y$ -coordinate $- \frac{1}{2}$ . Work out the $x$ -coordinates of $Q$ and $R$ .
[2 marks]
Answer $x = \dots\dots$ and $x = \dots\dots$
(c)
Hence write down the two solutions of $sin x = \frac{1}{2}$ for $36 0^{\circ} \leq x \leq 72 0^{\circ}$ .
[2 marks]
Answer $x = \dots\dots$ and $x = \dots\dots$
Total for D2: 6 marks
D3
Show that $tan^{2} 12 0^{\circ} - 4 sin 21 0^{\circ} cos 30 0^{\circ} = 4$ . You must show your working.
[4 marks]
D4
Which of these is equal to $cos 5 0^{\circ}$ ? Circle your answer.
$cos 13 0^{\circ} cos 23 0^{\circ} cos 31 0^{\circ} - cos 31 0^{\circ}$
[1 mark]
Answer
D5
$θ$ is an obtuse angle and $sin θ = \frac{5}{13}$ . Work out the exact values of $cos θ$ and $tan θ$ . Do not use a calculator.
[3 marks]
Answer $cos θ = \dots\dots$ $tan θ = \dots\dots$
D6
On the axes, sketch the graphs of $y = sin x$ and $y = cos x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . Hence find the range of values of $x$ for which $sin x > cos x$ . You must show your working.
[4 marks]
Answer $\dots\dots < x < \dots\dots$
D7
The graph shows $y = tan x$ for $- 36 0^{\circ} \leq x \leq 36 0^{\circ}$ and the line $y = k$ . The line meets the curve at the point where $x = 4 0^{\circ}$ . Work out the other values of $x$ , for $- 36 0^{\circ} \leq x \leq 36 0^{\circ}$ , where the line meets the curve.
[3 marks]
Answer $x = \dots\dots$

Extension

Stretch

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

E1
The curve $y = a cos x + b$ , where $a$ and $b$ are constants and $a > 0$ , has a greatest value of $7$ and a least value of $- 3$ . Work out $a$ and $b$ . Hence find the values of $x$ , for $0^{\circ} \leq x \leq 36 0^{\circ}$ , for which $y = 4.5$ . Do not use a calculator.
[4 marks]
Hint 1 · What to try
$cos x$ goes from $- 1$ to $1$ , so $y$ goes from $- a + b$ to $a + b$ .
Hint 2 · The first line
$a + b = 7$ and $- a + b = - 3$ .
Hint 3 · The full method
$a = 5$ and $b = 2$ , so $5 cos x + 2 = 4.5$ , $cos x = \frac{1}{2}$ , which gives $x = 6 0^{\circ}$ and $x = 360 - 60 = 30 0^{\circ}$ .
Answer $a = \dots$ $b = \dots$ $x = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the unit circle to find sin, cos and tan of any angle.
I can sketch the graphs of sine, cosine and tangent and give their key points.
I can use the symmetry of the graphs to find exact values and related angles.

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