AQA Level 2 Further Maths · Level 2 Certificate · Chapter 6: Geometry I · Lesson 4

Trigonometry in two dimensions

About 85 minutesCalculator allowed62 marks

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Date

Do Now

The first is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Write $\frac{12}{3}$ in the form $a 3$ , where $a$ is an integer. Do not use a calculator.
[2 marks]
Answer
A2
Last lesson
$P Q$ and $P R$ are tangents to a circle with centre $O$ , touching it at $Q$ and $R$ . Angle $QP R = 2 x$ . Prove that angle $QO R = 18 0^{\circ} - 2 x$ .
[2 marks]
A3
From chapter 5
Work out the distance between the points $(1, - 3)$ and $(5, 5)$ . Give your answer as a surd in its simplest form.
[2 marks]
Answer
A4
From chapter 4
Work out the value of $(\frac{27}{8})^{- \frac{2}{3}}$ . Give your answer as a fraction.
[2 marks]
Answer

Example, then your turn

Level 2

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

Example 1

A ladder leans against a vertical wall. The foot of the ladder is $2.3$ m from the wall. The ladder makes an angle of $6 8^{\circ}$ with the horizontal ground. Work out the length of the ladder. Give your answer to $3$ significant figures.

Not drawn accurately

Hint The

2.3

m is adjacent to the

6 8^{\circ}

and the ladder is the hypotenuse, so use cos. The unknown is on the bottom of the ratio.

B1
Work out the length $x$ . Give your answer to $3$ significant figures.
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ cm

Example 2

Triangle $A B C$ is isosceles with $A B = A C = 13$ cm and $B C = 18$ cm. Work out the size of angle $θ$ . Give your answer to $1$ decimal place.

Not drawn accurately

Hint The line of symmetry from

A

meets

B C

at its midpoint, at right angles. That makes a right-angled triangle with hypotenuse

13

and base

9

B2
Triangle $A B C$ is isosceles with $A B = A C = 11$ cm and angle $B A C = 4 8^{\circ}$ . Work out the length of $B C$ . Give your answer to $3$ significant figures.
[3 marks]
Not drawn accurately
Answer $B C = \dots\dots$ cm

Example 3

$F T$ is a vertical lighthouse $42$ m tall, standing on level ground. Two boats $P$ and $Q$ are on opposite sides of the lighthouse. From $T$ , the angle of depression of $P$ is $1 9^{\circ}$ and the angle of depression of $Q$ is $3 1^{\circ}$ . Work out the distance $P Q$ . Give your answer to $3$ significant figures.

Not drawn accurately

Hint An angle of depression is measured down from the horizontal. It equals the angle of elevation at the boat (alternate angles). Find

F P

and

F Q

separately.

B3
$C D$ is a building $20$ m tall on level ground. $D E$ is a vertical flagpole on top of it. From a point $G$ on the ground, the angle of elevation of $D$ is $3 8^{\circ}$ and the angle of elevation of $E$ is $5 2^{\circ}$ . Work out the height of the flagpole. Give your answer to $3$ significant figures.
[4 marks]
Not drawn accurately
Answer $\dots\dots$ m

Example 4

Work out the exact value of $x$ . Give your answer in the form $p + q 3$ , where $p$ and $q$ are integers. Do not use a calculator.

Not drawn accurately

Hint Half an equilateral triangle of side

2

has sides

1

3

and

2

, so

tan 3 0^{\circ} = \frac{1}{3}

B4
Work out the exact value of $x$ . Give your answer in the form $p + q 3$ , where $p$ and $q$ are integers. Do not use a calculator.
[3 marks]
Not drawn accurately
Answer $x = \dots\dots + \dots\dots 3$

Practice

Level 2

C1
$A B C D$ is a rectangle with $A B = 14.5$ cm and $B C = 6.2$ cm. Work out the obtuse angle between the diagonals $A C$ and $B D$ . Give your answer to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$
C2
$Q$ is $18$ km from $P$ on a bearing of $04 0^{\circ}$ . $R$ is $25$ km from $P$ . Angle $QP R = 9 0^{\circ}$ , with $R$ to the south-east of $P$ . Work out the distance $QR$ , to $3$ significant figures, and the bearing of $Q$ from $R$ , to the nearest degree.
[4 marks]
Not drawn accurately
Answer $QR = \dots\dots$ km bearing $\dots \dots^{\circ}$

C3
Work out the exact value of $\frac{sin 6 0 ^{\circ} + tan 3 0 ^{\circ}}{cos 3 0 ^{\circ}}$ . Do not use a calculator.
[3 marks]
Answer
C4
Prove that the area of an equilateral triangle with sides of length $a$ cm is $\frac{3}{4} a^{2}$ cm².
[3 marks]

Exam-style questions

AQA exam style

D1
Ravi says, " $sin 3 0^{\circ} + sin 3 0^{\circ} = sin 6 0^{\circ}$ ."
Show that Ravi is wrong. Do not use a calculator.
[2 marks]

D2
$C T$ is a vertical mast on horizontal ground. $A$ , $B$ and $C$ lie in a straight line, with $A B = 20$ m. The angle of elevation of $T$ from $A$ is $3 0^{\circ}$ and from $B$ is $4 5^{\circ}$ . Work out the exact height $C T$ . Give your answer in the form $p + q 3$ , where $p$ and $q$ are integers. Do not use a calculator. You must show your working.
[5 marks]
Not drawn accurately
Answer $C T = \dots\dots + \dots\dots 3$ m
D3
$A B C$ is a right-angled triangle. Angle $A B C = 9 0^{\circ}$ and $tan (angle B A C) = \frac{3}{4}$ . All lengths are in centimetres. Work out the perimeter of triangle $A B C$ .
[5 marks]
Not drawn accurately
Answer $\dots\dots$ cm
D4
$B F$ is a vertical post on horizontal ground. $A$ , $B$ and $C$ are on the ground in a straight line. $A B = 14$ m and the angle of elevation of $F$ from $A$ is $6 2^{\circ}$ . A straight cable $F C$ is $31$ m long.
Not drawn accurately
(a)
Work out the height of the post $B F$ .
[2 marks]
Answer $\dots\dots$ m
(b)
Hence work out the size of angle $F C B$ . Give your answer to $1$ decimal place.
[2 marks]
Answer $\dots \dots^{\circ}$
(c)
Work out the distance $A C$ .
[2 marks]
Answer $\dots\dots$ m
Total for D4: 6 marks
D5
$A$ and $B$ are points on a circle with centre $O$ and radius $7.5$ cm. Angle $A O B = 11 2^{\circ}$ .
(a)
Work out the length of the chord $A B$ .
[3 marks]
Answer $\dots\dots$ cm
(b)
Work out the shortest distance from $O$ to the chord $A B$ .
[2 marks]
Answer $\dots\dots$ cm
Total for D5: 5 marks

Extension

Stretch

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

E1
Triangle $A B C$ has a right angle at $C$ , $A C = 1$ and angle $A B C = 3 0^{\circ}$ . The side $C B$ is extended to $D$ so that $B D = B A$ . Use triangle $A C D$ to show that $tan 1 5^{\circ} = 2 - 3$ . Do not use a calculator.
[5 marks]
Not drawn accurately
Hint 1 · What to try
Find $A B$ and $C B$ using the exact values for $3 0^{\circ}$ . Then look at triangle $A B D$ : what kind of triangle is it?
Hint 2 · The first line
$A B = 2$ and $C B = 3$ . Triangle $A B D$ is isosceles, and angle $A B C = 3 0^{\circ}$ is an exterior angle of it, so angle $A D B = 1 5^{\circ}$ .
Hint 3 · The full method
$C D = 3 + 2$ , so $tan 1 5^{\circ} = \frac{1}{2 + 3}$ . Multiply the top and bottom by $2 - 3$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use sin, cos and tan to find a side or an angle, in context.
I can use angles of elevation and depression, and bearings.
I can use the exact values for $3 0^{\circ}$ , $4 5^{\circ}$ and $6 0^{\circ}$ without a calculator.

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