Further trigonometry 29 Opening problem A triangular property is bounded by two roads and a long, straight drain. Can you find: a the area of the property in m 2 and in hectares b the length of the drain boundary c the angle that the Johns Road boundary makes with the drain boundary? In Chapter 15 we introduced the unit circle, which is the circle with centre O(0, 0) and radius 1 unit. In that chapter we considered only the first quadrant of the circle, which corresponds to angles μ where 0 o 6 μ 6 90 o . We now consider the complete unit circle including all four quadrants. As P moves around the circle, the angle μ varies. The coordinates of P are defined as (cos μ, sin μ). THE UNIT CIRCLE [8.3] A 120° Johns Road Evans Road 277 m 324 m drain x y O q 1 A 1 -1 -1 P cos , ( ¡q ¡sin¡q) Contents: A The unit circle [8.3] B Area of a triangle using sine [8.6] C The sine rule [8.4] D The cosine rule [8.5] E Problem solving with the sine and cosine rules [8.4, 8.5, 8.7] F Trigonometry with compound shapes [8.1, 8.4, 8.5, 8.7] G Trigonometric graphs [3.2, 8.8] H Graphs of y = a sin(bx) and y = a cos(bx) [3.2, 3.3, 8.8]
26
Embed
Further trigonometry 29 - Haese Mathematics · F Trigonometry with compound shapes [8.1, 8.4, 8.5, ... two sides and the sine of the included angle. Example 4 Self Tutor ... 584 Further
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Further trigonometry
29
Opening problem
A triangular property is bounded by two roads and
a long, straight drain.
Can you find:
a the area of the property in m2 and in hectares
b the length of the drain boundary
c the angle that the Johns Road boundary makes
with the drain boundary?
In Chapter 15 we introduced the unit circle, which is the
circle with centre O(0, 0) and radius 1 unit. In that chapter
we considered only the first quadrant of the circle, which
corresponds to angles µ where 0o 6 µ 6 90o. We now consider
the complete unit circle including all four quadrants.
As P moves around the circle, the angle µ varies.
The coordinates of P are defined as (cos µ, sin µ).
THE UNIT CIRCLE [8.3]A
120°
Johns Road Evans Road277 m 324 m
drain
x
y
O�
1
A
1
��
��
P cos ,( ����sin��)
Contents:
A The unit circle [8.3]
B Area of a triangle using sine [8.6]
C The sine rule [8.4]D The cosine rule [8.5]E Problem solving with the
sine and cosine rules [8.4, 8.5, 8.7]
F Trigonometry with compound
shapes [8.1, 8.4, 8.5, 8.7]
G Trigonometric graphs [3.2, 8.8]
H Graphs of y = a sin(bx) and
y = a cos(bx) [3.2, 3.3, 8.8]
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\579IGCSE01_29.CDR Tuesday, 18 November 2008 11:07:41 AM PETER
Example 1 Self Tutor
a State the exact coordinates of:
i A ii B
b Find the coordinates of:
i A ii B
correct to 3 decimal places.
a i A(cos 152o, sin 152o) ii B(cos 297o, sin 297o)
b i A(¡0:883, 0:469) ii B(0:454, ¡0:891)
Example 2 Self Tutor
a Find the size of angle AOP marked with an
arrow.
b Find the coordinates of P using:
i the unit circle
ii symmetry in the y-axis.
c What can be deduced from b?
d Use c to simplify tan(180o ¡ µ).
y
x
( )����,(- )����,
(0 ),��
(0 ),��
10°
20°
30°
40°
50°
60°
70°80°100°
110°120°
130°
140°
150°
160°
170°
180°
190°
200°
210°
220°
230°
240°250°
260° 270° 280°290°
300°
310°
320°
330°
340°
350°
��.���.
���.
��.
Below is a unit circle diagram from which we can estimate trigonometric ratios.
x
y
O 1
1
��
��297°
B
A 152°152°
x
y
O
�
1
A
1
��
��
P
�
( )cos , sin���� ��
580 Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\580IGCSE01_29.CDR Monday, 27 October 2008 2:52:29 PM PETER
a angle AOP = (180o ¡ µ)
b i P is (cos(180o ¡ µ), sin(180o ¡ µ))
ii P is (¡ cos µ, sin µ)
c cos(180o ¡ µ) = ¡ cos µ and sin(180o ¡ µ) = sin µ
d tan(180o ¡ µ) =sin(180o ¡ µ)
cos(180o ¡ µ)
=sin µ
¡ cos µfusing cg
= ¡ tan µ
EXERCISE 29A.1
1 a
b Find the coordinates of P correct to 3 decimal places.
2 Use the unit circle diagram to find:
a sin 180o b cos 180o c sin 270o d cos 270o
e cos 360o f sin 360o g cos 450o h sin 450o
3 Use the unit circle diagram to estimate, to 2 decimal places:
a cos 50o b sin 50o c cos 110o d sin 110o
e sin 170o f cos 170o g sin 230o h cos 230o
i cos 320o j sin 320o k cos(¡30o) l sin(¡30o)
4 Check your answers to 3 using your calculator.
tan µ =sin µ
cos µ
5 a State the coordinates of point P.
b Find the coordinates of Q using:
i the unit circle
ii symmetry in the x-axis.
c What can be deduced from b?
d Use c to simplify tan(¡µ):
6 By considering a unit circle diagram like that in 5, show how to simplify
sin(180o + µ), cos(180o + µ), and tan(180o + µ).
Hint: Consider rotational symmetry.
x
y
O 1
1
��
��
231°
P
x
y
O 1
A
1
��
���
P
Q��
State the exact coordinates of P.
581Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\581IGCSE01_29.CDR Monday, 27 October 2008 2:52:32 PM PETER
IMPORTANT TRIGONOMETRIC RATIOS IN THE UNIT CIRCLE
In Chapter 15 we found the trigonometric ratios for the angles 0o, 30o, 45o, 60o and 90o.
µ cos µ sin µ tan µ
0o 1 0 0
30op32
12
1p3
45o 1p2
1p2
1
60o 12
p32
p3
90o 0 1 undefined
These angles correspond
to the points shown on
the first quadrant of the
unit circle:
We can use the symmetry of the unit circle to find
the coordinates of all points with angles that are
multiples of 30o and 45o.
For example, the point Q corresponding to an angle
of 120o is a reflection in the y-axis of point P with
angle 60o. ³¡1
2 ,p32
´.
Multiples of 30o Multiples of 45o
We can find the trigonometric ratios of these angles using the coordinates of the corresponding point on the
unit circle.
Example 3 Self Tutor
Use a unit circle diagram to find sin µ, cos µ and tan µ for:
a µ = 60o b µ = 150o c µ = 225o
x
y
(0, 1)
(1, 0)
(0, 1)�
( , 0)��
³12;p
32
´³p
32; 1
2
´
³p3
2;¡ 1
2
´³
12;¡
p3
2
´³¡ 1
2;¡
p3
2
´³¡
p3
2;¡ 1
2
´
³¡
p3
2; 1
2
´³¡ 1
2;p
32
´
120°120°
210°210° OO
Q has the negative -coordinate and the same -coordinate as P, so the coordinates of Q arex y
x
y
(0, 1)
(1, 0)
(0, 1)�
( 1, 0)�
³1p2; 1p
2
´
³1p2;¡ 1p
2
´³¡ 1p
2;¡ 1p
2
´
³¡ 1p
2; 1p
2
´
45°
135°135°
225°
315°315°
OO
O
30°30°45°45°
60°60°
y
x
³12 ;
p32
´³
1p2; 1p
2
´³p
32 ; 12
´
( )����,
( )����,
�� �
�
O
Q ,( )x y� P
y
x60° 60°
120°
³12 ;
p32
´
582 Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\582IGCSE01_29.CDR Monday, 27 October 2008 2:52:35 PM PETER
a
sin 60o =p32
cos 60o = 12
tan 60o =
p3212
=p3
b
sin 150o = 12
cos 150o = ¡p32
tan 150o =12
¡p32
= ¡ 1p3
c
sin 225o = ¡ 1p2
cos 225o = ¡ 1p2
tan 225o = 1
EXERCISE 29A.2
1 Use a unit circle to find sin µ, cos µ and tan µ for:
a µ = 30o b µ = 180o c µ = 135o d µ = 210o
e µ = 300o f µ = 270o g µ = 315o h µ = 240osin2 µ = (sin µ)2,
cos2 µ = (cos µ)2
and so on.2 Without using a calculator, find the exact values of:
a sin2 135o b cos2 120o c tan2 210o d cos3 330o
Check your answers using a calculator.
3 Use a unit circle diagram to find all angles between 0o and 360o which have:
a a sine of 12 b a cosine of
p32 c a sine of 1p
2
d a sine of ¡12 e a sine of ¡1 f a cosine of ¡
p32 .
Consider the acute angled triangle alongside, in which the sides
opposite angles A, B and C are labelled a, b and c respectively.
Area of triangle ABC = 12 £ AB £ CN = 1
2ch
But sinA =h
b
) h = b sinA
) area = 12c(b sinA) or 1
2bc sinA
If the altitudes from A and B were drawn, we could also show that
area = 12ac sinB = 1
2ab sinC. area = 1
2ab sinC is worth remembering.
AREA OF A TRIANGLE USING SINE [8.6]B
A B
C
A
ahb
Nc
C
B
y
x
³12 ,
p32
´
O
60°
y
x150°150°
³¡
p32 , 1
2
´O
y
x225°225°
³¡ 1p
2, ¡ 1p
2
´ O
583Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\583IGCSE01_29.CDR Monday, 27 October 2008 2:52:38 PM PETER
For the obtuse angled triangle ABC alongside:
Area of triangle ABC = 12 £ AB £ CN = 1
2ch
But sin(180o ¡A) =h
b) h = b sin(180o ¡A) = b sinA
) area of triangle ABC = 12cb sinA,
which is the same result as when A was acute.
Summary:
The area of a triangle is a half of the product of
two sides and the sine of the included angle.
Example 4 Self Tutor
Find the area of triangle ABC.
Area = 12ac sinB
= 12 £ 15£ 11£ sin 28o
¼ 38:7 cm2
EXERCISE 29B
1 Find the area of:
a b c
d e f
A
A
( )� � ���° A
N B
C
hb
a
c
included angleside
side
A
B C
11 cm
15 cm
28°
12 cm
13 cm
45°
28 km
82°
25 km
7.8 cm
112°
6.4 cm
1.65 m
78°
1.43 m12.2 cm
125°10.6 cm27 m84°32 m
584 Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\584IGCSE01_29.CDR Monday, 27 October 2008 2:52:41 PM PETER
2 Find the area of a parallelogram with sides 6:4 cm and 8:7 cm and one interior angle 64o.
3 If triangle ABC has area 150 cm2, find the value of x.
4 Triangle PQR has PbQR = µ. PQ = 10 m, QR = 12 m, and the area of the triangle is 30 m2.
Find the possible values of µ.
5 Triangle ABC has AB = 13 cm and BC = 17 cm, and its area is 73:4 cm2. Find the measure
of AbBC.
6 a Find the area of triangle ABC using:
i angle A ii angle C
b Hence, show thata
c=
sinA
sinC.
The sine rule is a set of equations which connects the lengths of the sides of any triangle with the sines of
the opposite angles.
The triangle does not have to be right angled for the sine rule to be used.
THE SINE RULE
In any triangle ABC with sides a, b and c units,
and opposite angles A, B and C respectively,
sinA
a=
sinB
b=
sinC
cor
a
sinA=
b
sinB=
c
sinC.
Proof: The area of any triangle ABC is given by 12bc sinA = 1
2ac sinB = 12ab sinC:
Dividing each expression by 12abc gives
sinA
a=
sinB
b=
sinC
c.
We use the sine rule when we are given:
² two sides and an angle not included between these sides, or
² two angles and a side.
THE SINE RULE [8.4]C
C
A
B
14 cm
x cm
75°
C
AA
C
B
ac
b
C
A B
b a
c
GEOMETRYPACKAGE
The sine rule is used to solve problems involving triangles when angles and sides
opposite those angles are to be related.
585Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\585IGCSE01_29.CDR Monday, 27 October 2008 2:52:44 PM PETER
FINDING SIDES
Example 5 Self Tutor
Find the length of side BC correct
to 2 decimal places:
Using the sine rule:BC
sin 113o=
18
sin 41o
) BC =18£ sin 113o
sin 41o
) BC ¼ 25:26
) BC is about 25:26 m long.
EXERCISE 29C.1
1 Find the value of x:
a b c
2 In triangle ABC find:
a a if A = 65o, B = 35o, b = 18 cm b b if A = 72o, C = 27o, c = 24 cm
c c if B = 25o, C = 42o, a = 7:2 cm.
FINDING ANGLES
The problem of finding angles using the sine rule is more complicated because there may be two possible
answers.
This ambiguous case may occur when we are given two
sides and one angle, where the angle is opposite the shorter
side.
It occurs because an equation of the form sin µ = b
produces answers of the form µ = sin¡1 b or
(180o ¡ sin¡1 b).
A
B
C
41°
113°18 m
15 cm
x cm
32°
46°x cm
48°
108°
9 cm6.3 km55°
84° x km
�� �
�
O
( )� �a b,
y
x
( )a b,�bb
��
586 Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\586IGCSE01_29.CDR Monday, 27 October 2008 2:52:46 PM PETER
Example 6 Self Tutor
Find, correct to 1 decimal place, the measure of angle C in triangle ABC if AC = 8 cm,
AB = 12 cm, and angle B measures 28o.
sinC
c=
sinB
bfsine ruleg
)sinC
12=
sin 28o
8
) sinC =12£ sin 28o
8
Now sin¡1
µ12£ sin 28o
8
¶¼ 44:8o
and since the angle at C could be acute or obtuse,
) C ¼ 44:8o or (180¡ 44:8)o
) C measures 44:8o if it is acute, or 135:2o if it is obtuse.
In this case there is insufficient information to determine the actual shape of the triangle.
The validity of the two answers in the above example can be demonstrated by a simple construction.
Step 1: Draw AB of length 12 cm and
construct an angle of 28o at B.
Step 2: From A, draw an arc of radius 8 cm.
Sometimes there is information given in the question which enables us to reject one of the answers.
Example 7 Self Tutor
Find the measure of angle L in triangle KLM given that LbKM measures 52o,
LM = 158 m, and KM = 128 m.
By the sine rule,sinL
128=
sin 52o
158
) sinL =128£ sin 52o
158
Now sin¡1
µ128£ sin 52o
158
¶¼ 39:7o
) L ¼ 39:7o or (180¡ 39:7)o ¼ 140:3o
But KM < LM, so we know angle L < angle K. Hence L ¼ 39:7o.
8 cm
8 cm
12 cm28°
135.2°
44.8°
A B
Cz
Cx
12 cm28°
A B
12 cm
8 cm
28°
A B
C
K M
L
128 m
158 m
52°
587Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\587IGCSE01_29.CDR Monday, 27 October 2008 2:52:49 PM PETER
EXERCISE 29C.2
1 Find the value of µ:
a b c
2 Solve for x:
a b c
d e f
3 In triangle ABC, find the measure of:
a angle A if a = 12:6 cm, b = 15:1 cm and AbBC = 65o
b angle B if b = 38:4 cm, c = 27:6 cm and AbCB = 43o
c angle C if a = 5:5 km, c = 4:1 km and BbAC = 71o.
4 In triangle ABC, angle A = 100o, angle C = 21o, and AB = 6:8 cm.
Find the length of side AC.
5 In triangle PQR, angle Q = 98o, PR = 22 cm, and PQ = 15 cm.
Find the size of angle R.
THE COSINE RULE
In any triangle ABC with sides a, b and c units and opposite angles A, B and C respectively,
a2 = b2 + c2 ¡ 2bc cosA
b2 = a2 + c2 ¡ 2ac cosB
c2 = a2 + b2 ¡ 2ab cosC.
THE COSINE RULE [8.5]D
14.8 m 17.5 m
38° �°
29 cm
54°
35 cm
�°
2.4 km6.4 km
15°�°
A B
C
ab
c
There may be twopossible solutions.
A sketch mayhelp to find them.
6 m
8 m
x°
20°
120°
x cm
9 cm
25°
x km
6 km
x°
5 m
50°
7 m
110°
x m
10 m10 cm
x° 12 cm
35°
25° 30°
588 Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\588IGCSE01_29.CDR Monday, 27 October 2008 2:52:52 PM PETER
Proof (for a triangle with acute angle A):
Consider triangle ABC shown.
Using Pythagoras’ theorem, we find
b2 = h2 + x2, so h2 = b2 ¡ x2
and a2 = h2 + (c ¡ x)2
Thus, a2 = (b2 ¡ x2) + (c ¡ x)2
) a2 = b2 ¡ x2 + c2 ¡ 2cx + x2
) a2 = b2 + c2 ¡ 2cx :::::: (1)
But in ¢ACN, cosA =x
band so x = b cosA
So, in (1), a2 = b2 + c2 ¡ 2bc cosA
Similarly, we can show the other two equations to be true.
Challenge: Prove the Cosine Rule a2 = b2 + c2 ¡ 2bc cosA, in the case where A is an obtuse angle.
You will need to use cos(180o ¡ µ) = ¡ cos µ:
We use the cosine rule when we are given:
² two sides and the included angle between them, or
² three sides.
Useful rearrangements of the cosine rule are:
cosA =b2 + c2 ¡ a2
2bc, cosB =
a2 + c2 ¡ b2
2ac, cosC =
a2 + b2 ¡ c2
2ab
They can be used if we are given all three side lengths of a triangle.
Example 8 Self Tutor
Find, correct to 2 decimal places,
the length of BC.
By the cosine rule:
a2 = b2 + c2 ¡ 2bc cosA
) a =p
122 + 102 ¡ 2 £ 12 £ 10 £ cos 38o
) a ¼ 7:41
) BC is 7:41 m in length.
A B
C
A
b a
x c x���
h
N
A C
B
10 m
38°
12 m
A C
B
10 m
38°
12 m
a m
589Further trigonometry (Chapter 29)
GEOMETRYPACKAGE
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\589IGCSE01_29.CDR Friday, 31 October 2008 9:54:56 AM TROY
C
BA
98°
21 cm
15 cm
P
R
Q
4.8 km
38°
6.7 km
K
L M
10.3 m
72°
14.8 m
A B
C
9 m 11 m
14 m
Example 9 Self Tutor
Find the size of AbBC in the given figure.
Give your answer correct to 1 decimal place.
cosB =a2 + c2 ¡ b2
2ac
) cosB =112 + 82 ¡ 92
2 £ 11 £ 8
) B = cos¡1
µ112 + 82 ¡ 92
2 £ 11 £ 8
¶) B ¼ 53:8o
So, AbBC measures about 53:8o.
EXERCISE 29D
1 Find the value of x in:
a b c
d e f
2 Find the length of the remaining side in the given triangle:
a b c
3 Find the measure of all angles of:
A
B C
B
8 m 9 m
11 m
5 m80°
x m
6 m10 cm
x cm
140°
11 cm
3.8 km
100°5.3 km
x km
3 m
4 m
2 mx°
9 cm12 cm
x°
7 cm
8 km
11 km
x°
14 km
590 Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\590IGCSE01_29.CDR Friday, 31 October 2008 9:51:46 AM TROY
4 Find:
a the smallest angle of a triangle with sides 9 cm, 11 cm and 13 cm
b the largest angle of a triangle with sides 3 cm, 5 cm and 7 cm.
5 a Use the cosine rule in triangle BCM to find cos µ in terms
of a, c and m.
b Use the cosine rule in triangle ACM to find cos(180o¡ µ)
in terms of b, c and m:
c Use the fact that cos(180o ¡ µ) = ¡ cos µ to prove
Apollonius’ median theorem:
a2 + b2 = 2m2 + 2c2.
d Hence find x in the following:
i ii
6 In triangle ABC, AB = 10 cm, AC = 9 cm and AbBC = 60o. Let BC = x cm.
a Use the cosine rule to show that x is a solution of x2 ¡ 10x+ 19 = 0.
b Solve the above equation for x.
c Use a scale diagram and a compass to explain why there are two possible values of x.
7 Find, correct to 3 significant figures, the area of:
a b
Whenever there is a choice between using the sine rule or the cosine rule, always use the cosine rule to
avoid the ambiguous case.
Example 10 Self Tutor
An aircraft flies 74 km on a bearing 038o and then 63 km on a bearing 160o.
Find the distance of the aircraft from its starting point.
PROBLEM SOLVING WITH THESINE AND COSINE RULES
E
C B
A
b
m
a
c
c
�M
� � ����°
12 cm9 cm
x cm
5 cm
8 m
8 m
10 m
x m
2 cm
3 cm
4 cm
4 cm
5 cm
6 cm
591Further trigonometry (Chapter 29)
[8.4, 8.5, 8.7]
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\591IGCSE01_29.CDR Tuesday, 18 November 2008 11:08:11 AM PETER
By the cosine rule,
b2 = a2 + c2 ¡ 2ac cosB
) b2 = 632 + 742 ¡ 2£ 63£ 74£ cos 58o
) b2 ¼ 4504:03
) b ¼ 67:1
) the aircraft is 67:1 km from its starting point.
EXERCISE 29E
1 Two farm houses A and B are 10:3 km apart. A third farm house C is located such that BbAC = 83o
and AbBC = 59o. How far is C from A?
2 A roadway is horizontal for 524 m from A to B,
followed by a 23o incline 786 m long from B to C.
How far is it directly from A to C?
3 Towns A, B and C are located such that BbAC = 50o and B is twice as far from C as A is from C.
Find the measure of BbCA.
4 Hazel’s property is triangular with dimensions as shown
in the figure.
a Find the measure of the angle at A, correct to 2decimal places.
b Hence, find the area of her property correct to the
nearest hectare.
5 An aeroplane flies from Geneva on a bearing of 031o for 200 km. It then changes course and flies for
140 km on a bearing of 075o. Find:
a the distance of Geneva from the aeroplane
b the bearing of Geneva from the aeroplane.
6 A ship sails northeast for 20 km and then changes direction, sailing on a bearing of 250o for 12 km.
Find:
a the distance of the ship from its starting position
b the bearing it must take to return directly to its starting position.
7 An orienteer runs for 450 m then turns through an angle of 32o and runs for another 600 m. How far
is she from her starting point?
8 A yacht sails 6 km on a bearing 127o and then 4 km on a bearing 053o. Find the distance and bearing
of the yacht from its starting point.
9 Mount X is 9 km from Mount Y on a bearing 146o. Mount Z is 14 km away from Mount X and on a
bearing 072o from Mount Y. Find the bearing of X from Z.
10 A parallelogram has sides of length 8 cm and 12 cm. Given that one of its angles is 65o, find the
lengths of its diagonals.
11 Calculate the length of a side of a regular pentagon whose vertices lie on a circle with radius 12 cm.
142° B 160°
63 km
C
A
b km
38°
58°74 km
N
N
AB
C
hill
524 m
23°786 m
A
B
C
314 m238 m
407 m
592 Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\592IGCSE01_29.CDR Monday, 27 October 2008 2:53:04 PM PETER
12 X is 20 km north of Y. A mobile telephone mast M is to be placed 15 km from Y so the bearing of M
from X is 140o.
a Draw a sketch to show the two possible positions where the mast could be placed.
b Calculate the distance of each of these positions from X.
13 Bushwalkers leave point P and walk in the direction 238o for 11:3 km to point Q. At Q they change
direction to 107o and walk for 18:9 km to point R. How far is R from the starting point P?
14 David’s garden plot is in the shape of a quadrilateral. If the corner
points are A, B, C and D then the angles at A and C are 120o
and 60o respectively. AD = 16 m, BC = 25 m, and DC is 5 m
longer than AB. A fence runs around the entire boundary of the
plot. How long is the fence?
Example 11 Self Tutor
AD is a vertical mast and CE is a vertical flagpole.
Angle ABD is 30o and angle EBC is 50o.
Calculate:
a the length of DE
b the size of angle EDB.
a angle DBE = 180o ¡ 30o ¡ 50o
= 100o
) DE2 = 92 + 12 ¡ 2(9)(12) cos 100o
fby the Cosine Ruleg) DE ¼ 16:202 m
) DE ¼ 16:2 m
b
Using the sine rule:sin µ
12¼ sin 100o
16:202
) sin µ ¼ 12£ sin 100o
16:202
) µ ¼ 46:8o
) angle EDB is about 46:8o:
TRIGONOMETRY WITH COMPOUND SHAPESF
A
B
CD
120°
16 m
25 m
60°
9 m
12 m
E
D
100°
�
16.202 m
B
50°
9 m
12 m
A B C
E
D
30°
2
593Further trigonometry (Chapter 29)
[8.1, 8.4, 8.5, 8.7]
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\593IGCSE01_29.CDR Tuesday, 18 November 2008 11:08:38 AM PETER
EXERCISE 29F
1 A 40 m high tower is 8 m wide. Two students A and B are
on opposite sides of the top of the tower. They measure the
angles of depression to their friends at C and D to be 54o and
43o respectively. How far are C and D apart if A, B, C and
D are all in the same plane?
2 The area of triangle ABD is 33:6 m2.
Find the length of CD.
3 Find x:
4 A stormwater drain is to have the shape shown. Determine
the angle the left hand side makes with the bottom of the
drain.
5
a What is the size of AbTB?
b Find the distance from A to T.
c Find the distance from B to T.
d Find the height of the mountain.
e Use the given figure to show that
d = h
µ1
tan µ¡ 1
tanÁ
¶:
f Use e to check your answer to d.
6 Find x and y in the given figure.
A B
T
1200 m37° 41°
� �
d x
h
54° 43°A B
C D8 m
40 m
10 cm x cm
65° 48°
2 m
100°
5 m
3 m
�°
From points A and B at sea, the angles of elevation to
the top of the mountain T are and respectively.
A and B are m apart.
37 411200
o o
95°
118°
30°
y m
x m
22 m
B 7 m
10 m
9 m CA
D
594 Further trigonometry (Chapter 29)
IGCSE01
magentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IGCSE01\IG01_29\594IGCSE01_29.CDR Monday, 27 October 2008 2:53:10 PM PETER
7 Jane and Peter are considering buying a block of land. The
land agent supplies them with the given accurate sketch. Find
the area of the property, giving your answer in:
a m2 b hectares.
8 In the given plan view, AC = 12 m, angle BAC = 60o,
and angle ABC = 40o. D is a post 6 m from corner B,
E is another post, and BDE is a lawn of area 13:5 m2.
a Calculate the length of DC.
b Calculate the length of BE.
c Find the area of ACDE.
GRAPHS FROM THE UNIT CIRCLE
The diagram alongside gives the y-coordinates for all points on
the unit circle at intervals of 30o.
A table for y = sin µ can be constructed from these values: