STUDENT MATERIALS

CONTENTS

TrigonometryIntroduction: Sine, Cosine and Tangents of non-acute angles A. Area of a triangleB. Sine RuleC. Cosine Rule

Checkup

Simultaneous EquationsA. Construction of FormulaeB. Solving Simultaneous Equations (Graphically)C. Solving Simultaneous Equations (Algebraically)

Checkup

Specimen Assessment Questions

Answers

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 1

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 2

TRIGONOMETRY

By the end of this set of exercises, you should be able to

(a) calculate the area of a triangle using trigonometry

(b) solve problems using Sine and Cosine rules.

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 3

TRIGONOMETRY

Introduction: Sine, Cosine and Tangent Graphs

Exercise 1A

1. The Sine Graph (a) Make a copy of this table and use your calculator to help fill it in, giving each

answer correct to 2 decimal places.

(b) Use a piece of 2 mm graph paper to draw a set of axes as illustrated below.

(c) Plot as accurately as possible the 21 points from your table.

(d) Join them up smoothly to create the graph of the function y = sin x.

2. Repeat question 1 (a) to (d) for the function y = cos x

3. Repeat for the graph of y = tan x (a different scale will be required for the vertical axis).(These graphs will be studied later).

Sine, Cosine and Tangents of angles other than acute angles

Exercise 1B

1. Use your calculator to find the following trigonometric ratios.Give each answer correct to 3 decimal places.(a) sin 25 (b) cos 95 (c) tan 107 (d) sin 200(e) cos 315 (f) tan 181 (g) cos 240 (h) sin 330(i) tan 225 (j) sin 300 (k) tan 315 (l) cos 500(m) tan (75) (n) cos (200) (o) sin 360 (p) cos 360

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 4

x 0 20 40 60 80 90 100 120 140 160 180sin x 000 034 064 087 098 100 . . . . . . . . . . . . . . .

x 200 220 240 260 270 280 300 320 340 360sin x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 0 1 8 0 2 7 0 3 6 00

1

1

A . Area of a Triangle using Trigonometry.

Exercise 2

1. In this question you are being asked to calculate the area of triangle ABC, using two methods.

Method 1 (a) Use basic right angledtrigonometry on triangleABP to calculate theheight BP (= h cm).

(b) Now use the formulaArea = 1/2 (base x height)to calculate the area of DABC.

Method 2 Use the formula:

Area = 1/2 b c sin A with b = 12 cm, c = 10 cm and angle A = 72

to calculate the area of triangle ABC.Did you obtain the same answer? Which method was the faster?

2. Use the formula Area = 1/2 a b sin C to calculate the areas of the following six triangles:(Give all answers correct to 1 decimal place).

(a) (b)

(c) (d)

(e) (f)

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 5

A

h cm10 cm

P C

B

72

12 cm

A

B

C

6 cm

7 cm42

Y

X

Z

13 cm

16 cm

105

TS

R

32 cm

21 cm

128

E

D

F

16 cm 16 cm

34

N

M

L

92 cm

85 cm

52

RQ

P

15 cm

12 cm

39

3. Calculate the areas of the following two triangles:(a) (b)

What do you notice?

4. Calculate the areas of the following two triangles:(a) (b)

What do you notice? Can you explain your answers to questions 3 and 4?

5. Shown is a sketch of Farmer Gilestriangular field.Calculate its area in square metres.

6. Calculate the area of this pentagon:

7. Calculate the areas of the following two parallelograms: (a) (b)

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 6

10 cm

8 cm

40

10 cm

8 cm

140

65 cm

7 cm

53 65 cm

7 cm

127

618 cm

10 cm

10 cm

30

15 cm

14 cm

17 cm

65

4 cm

8 cm

125

65 m

52 m

43

B . Sine Rule.

Exercise 3

In this exercise, give all answers correct to 1 decimal place.

1. Copy and complete the following:

= = ( )

=

=> a = = cm

2. Use the Sine Rule in each of the following to calculate the size of the side marked x cm.

(a) (b)

(c) (d)

(e) (f)

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 7

75 cma cm

61

39A

B

C

aSin A

75 x Sin 61Sin 39

75Sin 39

aSin 61

bSin B

cSin C

J

I

K24

19 cm

110

x cm

A

C

B

25 cm x cm

4272

CA

B

19 cm

84 47

x cm

TS

R

45 cm

35

120

x cm

P

Q

R60

86 cm

50

x cm

D

E

F9 cm

70

40

x cmx cm

3. (a) Write down the size of PQR.

(b) Use the Sine rule to calculatethe length of the line QR.

4. In each of the following, calculate the sizeof the third angle first before attempting to calculate the length of the side marked x cm.(a) (b) (c)

5. Copy and complete:

= = ( )

=

=> 8 Sin x = 10 Sin 42

=> Sin x = = 0 .....

=> x =

6. Use the Sine Rule in each of the following to calculate the size of the angle marked x .(a) (b)

(c) (d)

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 8

Q

9 cm

80

60

R

P

B

65 cm

52 61

CA

x cm

M

PN 104 cm

59

72

x cm

J

KF

26 cm42

109

x cm

8 cm10 cm

x

42A

B

C

aSin A

10 Sin 428

8Sin 42

10Sin x

bSin B

cSin C

12 cm10 cm

x

47A

B

C

30 cm

25 cmx

54

Q

P

R

12 cm

10 cmx 112

Y

X

Z

32 cm

84 cm

x

75

L

M

N

7. The diagram shows a roof truss.Calculate the size of the angle markedx between the wooden supports.

8. H.M.S. Nautilus lies East of H.M.S. Unicorn.The diagram shows where an enemy submarine is in relation to the two ships.Calculate how far the submarine is from H.M.S.Nautilus.

9. This is the metal frame used to support andhold a childs swing.It is in the shape of an isosceles triangle.

(a) Calculate the size of ABC.

(b) Use the Sine rule to calculate howfar apart points B and C are.(Answers to 2 decimal places)

(c) Draw a vertical line through A, creating two right angled trianglesand use right angled trigonometry to check your answer to part (b).

10. Calculate the size of the angles marked x , y and z . (careful!)

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 9

Unicorn Nautilus

35 km

32 51

N N

A

30

B C

28 m 28 m

A

x

B

C

10 cm9 cm

65 P

y

Q

R

106 cm

61 cm

108

U

z

V

W

98 cm77 cm

42

34

32 m 21 m

x

C . Cosine Rule

Exercise 4A

1. Copy and complete the following:

a2 = b 2 + c 2 (2bc cos A)

=> x 2 = 7 2 + 8 2 (2 x 7 x 8 x cos 25)

=> x 2 = ... + ... (.....)

=> x 2 = ......

=> x =

2. Use the Cosine rule to calculate the size of each side marked x cm here.(a) (b)

(c) (d)

(e) (f)

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 10

C

7 cm

25

B

A

8 cm

x cm

43 cm

x cm

29 cm

51

J

KI

10 cm

x cm 9 cm

34

B

CA15 cm

x cm

12 cm

63

R

Q

P

87 cm

x cm 92 cm

47

M

NL

75 cm

x cm

75 cm

34

G

FE

18 cm

x cm20 cm

15W

V

Y

3. Copy and complete the following:

a2 = b 2 + c 2 (2bc cos A)

=> x 2 = 8 2 + 6 2 (2 x 8 x 6 x cos 110)

=> x 2 = ... + ... (96 x (0342..))

=> x 2 = ...... (3283..)

=> x 2 = ...... + 3283..

=> x 2 = ......

=> x =

4. Calculate the lengths of the sides marked x cm.(a) (b) (c)

5. A farmer owns a piece of fenced land which istriangular in shape.Calculate the length of the third side and thenwrite down the perimeter of the field.

6. Two ships leave Peterborough harbour at 1300. The Nightingale sails at 20 miles per hour on a bearing 042. The Mayflower II sails at 25 miles per hour on a bearing 087.

(a) Calculate the size of NMP.

(b) How far apart will the 2 ships be after 1 hour?

(c) How far apart will they be at 1600?

Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 11

C

8 cm

110

x cm

B6 cmA

(note)

C9 cm

95

x cmB

7 cm

A

R

15 cm120

x cm

Q 17 cm

PV48 cm

131

x cm

U