35:The Sine Rule35:The Sine Rule

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

The Sine Rule

Triangles that aren’t Right Angled

To find unknown sides and angles in non-right angled triangles we can use one or both of 2 rules: • the sine

rule• the cosine rule

The next few slides prove the sine rule. The cosine rule is on the next presentation.

You do not need to learn the proof.

The Sine Rule

a, b and c are the sides opposite angles A, B and C

A B

C

b a

c

ABC is a scalene triangle

The Sine Rule

The Sine Rule

A B

C Draw the perpendicular, h, from C to BA.

N

h b a

c

In ,ACNΔ

ABC is a scalene triangle

The Sine Rule

A

h b a

c

Asin

b a

c

C

B

In ,ACNΔ

ABC is a scalene triangle

N

The Sine Rule

A B

h b a

c

C In ,ACNΔ

hAb sin

ABC is a scalene triangle

In ,BCNΔ

N

Asinb

h

The Sine Rule

Bsin

A B N

h b a

c A B N

h b a

c

C In ,ACNΔ

In ,BCNΔ

ABC is a scalene triangle

hAb sin

Asinb

h

The Sine Rule

hBa sin

In ,ACNΔ

In ,BCNΔ

ABC is a scalene triangle

hAb sin

A B N

h b a

c A B

h b a

c

C

a

hBsin

Asinb

h

The Sine Rule

BaAb sinsin

so, Ab sin Ba sinandh h

A B

C

hb a

c

The Sine Rule

BaAb sinsin

A B

C

b a

c B

b

A

a

sinsin

so, Ab sin Ba sinandh h

angle) oppositesin(

side

angle) oppositesin(

side

b

B

a

A sinsin

The Sine Rule

. . . can be turned so that BC is the base. CbBch sinsin

C

A B

b a

c

A

B C

b

a

c

The triangle ABC . . .

We would then get

hC

c

B

b

sinsin

c

C

b

B sinsin

The Sine Rule

c

C

b

B sinsinSo,

b

B

a

A sinsin

c

C

b

B sinsin

c

C

b

B

a

A sinsinsin

We now have

So,

and

C

c

B

b

A

a

sinsinsin

The Sine Rule

The sine rule can be used in a triangle when we know• One side and its opposite angle,

plus• One more side or angle

sinsin

P Q

q p

Tip: We need one complete “pair” to use the sine rule.

e.g. Suppose we know p, q and angle Q in triangle PQR

The angle or side that we can find is the one that completes another pair.

The Sine Rule

sinsin

A B

b a Solution:

Usesin

sin AB a

b 12

62sin10sin A

We don’t need the 3rd part of the rule

7358.0sin A

e.g. 1 In the triangle ABC, find the size of angles A and C.

A B

C

12 10

62

67044762180 C

447A (3 s.f.)

The Sine Rule

Solution: As the unknown is a side, we “flip” the sine rule over. The unknown side is then

at the “top”. z

sin

sin Z y

Y

29sin

55sin13 z 022 z ( 3 s.f.)

sinsin

z y

Z Y

e.g. 2 In the triangle XYZ, find the length XY.

Y Z

X

13

5529

z

The Sine RuleSUMMARY The sine rule can be used in a triangle

when we know• One side and its opposite angle, plus• One more side or angle

If 2 sides and 1 angle are known we use:

If 1 side and 2 angles are known we use:

We write the sine rule so that the unknown angle or side is on the left of the equation

B

b

A sinsina

b

B

a

sinsinA

The Sine RuleExercises

1. In triangle ABC, b = cm, c = cm and angle C = . Find the size of angles A and B.

1102463

Solution:

sinsin

B C

c b

24

110sin63sin

B

753 B 316 A

2. In triangle PQR, PQ = 23 cm, angle R = and angle P = . Find the size of side QR.

4217

BA

C

a 63

24

110

The Sine Rule

42sin

17sin23 p

010 p ( 3 s.f.)

2. In triangle PQR, PQ = 23 cm, angle R = and angle P = . Find the size of QR.

4217

sinsin

p r

P R

Solution:

23P Q

R

p

42

17

Exercises

The Sine Rule

The following may be left out if time is an issue

The Sine Rule

e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm and angle P = .

37

Drawing side r and angle P, we have:

P Q7.2

37

R1

5 This is one possible complete triangle.

If an unknown angle is opposite the longest side, 2 triangles may be possible: one will have an angle greater than90

The Sine Rule

This is the other.

P Q

R1

7.2

5

37

5

R2

e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm and angle P = .

37

The Sine Rule

This is the other.

P Q

R1

7.2

5

37

5

R2

The 2 possible values of R are connected since

x

xy

xR 1

• Triangle is isosceles

21 RQR

180 yx•

e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm and angle P = .

37

The Sine Rule

This is the other. The 2 possible values of R are connected since

xR 1

• Triangle is isosceles

21 RQR

180 yx•

e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm and angle P = .

37

P Q

R1

7.2

5

37

5

R2

x

xy180so, 2R1R

The calculator will give the acute angle ( < ).We subtract from to find the other possibility.180

90

The Sine Rule

e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm andangle P = . Find 2 possible values of angle R and the corresponding values of angle Q. Give the answers correct to the nearest degree.

37

Solution:

sinsin

R P p r

5

37sin27sin

R

p

PrR

sinsin

86660sin R601 R or

120601802 R8360371801 Q23120371802 Q

601 R1202 R

The Sine Rule

or:

We have either:

60R 83Qand

120R 23Qand

120

23

PQ27.2

37

5

R2

P

83

60

Q1

R1

7.2

5

37

The Sine RuleSUMMARY

If the sine rule is used to find the angle opposite the longest side of a triangle, 2 values may be possible.

Use each value to find 2 possible values for the 3rd angle.

Use the sine rule and a calculator to find 1 value. This will be an acute angle ( less than ).

90

Subtract from to find the other possibility.

180

The Sine RuleExercise

Solution:

sinsin

C B b c

91420sin C12

47sin15sin C

96647166180 A

119479113180 A

166C9113 C

Find 2 possible values of angle ACB in triangle ABC if AB = 15 cm, AC = 12 cm and angle B = . Sketch the triangles obtained.

47

9113166 CC or

The Sine Rule

966A 119A

AB = 15 cm, AC = 12 cm, angle B = 47

A B15

166

C

12

47966

9113

A B15

119

C12

47

166C(i)

9113 C(ii)

Exercise

The Sine Rule

The Sine Rule

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

The Sine Rule

c

C

b

B

a

A sinsinsin

a, b and c are the sides opposite angles A, B and C

A B

C

b a

c

ABC is a scalene triangle

The Sine Rule

The Sine Rule

The sine rule can be used in a triangle when we know• One side and its opposite angle,

plus• One more side or angle

If 2 sides and 1 angle are known ( a, b and B ) we use:

If 1 side and 2 angles are known ( A, b and B ) we use:

We write the sine rule so that the unknown angle or side is on the left of the equation

B

b

A sinsina

b

B

a

sinsinA

The Sine Rule

We don’t need the 3rd part of the rule

sinsin

A B

b a Solution:

Use

sinsin A

B a

b 12

62sin10sin A

7358.0sin A

67044762180 C

447A (3 s.f.)

e.g. 1 In the triangle ABC, find the size of angles A and C.

A B

C

12 10

62

The Sine Rule

Solution: As the unknown is a side, we “flip” the sine rule over. The unknown side is then

at the “top”.

zsin

sin Z y

Y

29sin

55sin13 z 022 z ( 3 s.f.)

sinsin

z y

Z Y

e.g. 2 In the triangle XYZ, find the length XY.

Y Z

X

13

5529

z

The Sine RuleIf an unknown angle is opposite the longest side, 2 triangles may be possible: one will have an angle greater than90

e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm andangle P = . Find 2 possible values of angle R and the corresponding values of angle Q. Give the answers correct to the nearest degree.

37

Solution:

sinsin

R P p r

5

37sin27sin

R

p

PrR

sinsin

86660sin R601 R or

120601802 R8360371801 Q23120371802 Q

601 R1202 R