36: The Cosine Rule36: The Cosine Rule

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

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

The Cosine Rule

Module C2

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The Cosine Rule

The cosine rule is used to find sides and angles of a scalene triangle when

• 2 sides and the angle formed by them are known, or

• all 3 sides are known

In both these cases, we don’t know a pair of side and opposite angle so the sine rule cannot be used.

We will now prove the cosine rule but you do not need to learn the proof.

The Cosine Rule

In the triangle ABC, draw the perpendicular, h, from C to AB.

Proof of the Cosine Rule

N

h

Let AN = x. Then, NB = c x.

x c - x Using Pythagoras’ theorem:In triangle

ANC,In triangle BNC,So, )2()( 2222 xcaxb

From triangle ANC,

)1(cos Abxb

xA cos

A B

C

b a

c

h2 22 xb

h2 22 )( xca

The Cosine Rule

)2( 22222 xcxcaxb

222 cos2 aAbccb

Acbcab cos2222

)2()( 2222 xcaxbWe have

Simplifying:22222 2 xcxcaxb

cxcab 2222

Abccba cos2222

)cos( Abx Substituting for x from equation (1),Rearranging:

Proof of the Cosine Rule

The Cosine Rule

• The letters can be switched to find any side provided it is opposite the given angle.

Abccba cos2222

The Cosine Rule for triangle ABC

• We use this arrangement when 2 sides and the angle formed by them are known.

The Cosine Rule

cos222 bccb 2a A

• The letters can be switched to find any side provided it is opposite the given angle.

The Cosine Rule for triangle ABC

• We use this arrangement when 2 sides and the angle formed by them are known.

• If we want to find an angle, we use the sine rule after we have used the cosine rule.

The Cosine Rule

Cbaabc cos2222 19

e.g. Find side c and angle B in the triangle ABC

A

B C

15 c

30

30cos)19)(15(21915 222 c

( 3 s.f.)619c

Solution: Use the Cosine rule

The Sine rule: c

C

b

B sinsin

619

30sin15sin

B

351B ( 3 s.f.)

Tip: Do the whole calculation in one go on

your calculator. It avoids errors!

Tip: Leave the answer on your calculator as it will be needed to find angle B

a

b

The Cosine RuleExercise

Pqrrqp cos2222

7

1. Find p in the triangle PQR

P R

Q

6

p

120

120cos)6)(7(267 222 p

( 3 s.f.)311p

Solution:

The Cosine Rule

. . . belongs to the side opposite the angle we are finding

The 2nd form of the Cosine Rule

Abccba cos2222

222cos2 acbAbc

We know that

Rearranging,

bc

acbA

2cos

222

We use this form to find any angle of a triangle when we know all 3 sides.

The minus sign . . .

The Cosine Rule

The Cosine Rule

Xcos

Solution: Use the Cosine Rule

6

e.g. 1 Find angle X in triangle XYZ

Y

Z 8

4

X

029X

8750cos X)6)(8(2

468 222

yz

xzyX

2cos

222

The Cosine Rule

bc

acbA

2cos

222 Acos

Solution: Let’s find A first

6e.g. 2 Find all the

angles in triangle ABC B

A 9

5

C

938A77780cos A)5)(9(2

659 222

We can now use the Cosine rule again or switch to the Sine rule. If we use the Sine rule, we must avoid the largest angle ( opposite the longest side ) as we don’t know whether it is less than or greater than . 90

The Cosine Rule

OR: Using the Sine rule for C :631C

6

938sin5sin

C

EITHER: Using the Cosine rule for B or C: 5109 B

)6)(5(2

965cos

222 B 3330cos B

a

A

c

C sinsin

e.g.

6

B

A 9

5

C

938A

6319385109180 C Then

5109938631180 B Then

The Cosine Rule

The Cosine Rule

SUMMARY

Abccba cos2222

• In a triangle that isn’t right angled, if we know 2 sides and the angle formed by the 2 sides, we use

• If we know 3 sides, we use

bc

acbA

2cos

222

to find the 3rd side.

to find any angle.

The Cosine RuleExercise

7

1. Find all the angles in the triangle XYZ giving the answers to 1 decimal place.

Y Z

X

4 9

)9)(4(2

794cos

222 X 248X

Solution: Use the Cosine rule for any angle. e.g.

yz

xzyX

2cos

222

)9)(7(2

497cos

222 Y 225Y

(1 d. p.)6106248225180 Zxz

yzxY

2cos

222

The Cosine Rule

The Cosine 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 Cosine Rule

The cosine rule is used to find sides and angles of a scalene triangle when

• 2 sides and the angle formed by them are known, or

• all 3 sides are known

In both these cases, we don’t know a side and its opposite angle so the sine rule cannot be used.

The Cosine Rule

• The letters can be switched to find any side provided it is opposite the given angle.

Abccba cos2222

The Cosine Rule for triangle ABC

• We use this arrangement when 2 sides and the angle formed by them are known.

• If we want to find an angle, we use the sine rule after we have used the cosine rule.

The Cosine Rule

Cbaabc cos2222

19

e.g. Find side c and angle B in the triangle ABC

A

B C

15 c

30

30cos)19)(15(21915 222 c

( 3 s.f.)619c

Solution: Use the Cosine rule

The Sine rule: c

C

b

B sinsin

619

30sin15sin

B

351B ( 3 s.f.)

Tip: Do the whole calculation in one go on your calculator and leave your answer so it can be used to find B.

a

b

The Cosine Rule

The 2nd form of the Cosine Rule

Abccba cos2222

222cos2 acbAbc

We know that

Rearranging,

bc

acbA

2cos

222

We use this form to find any angle of a triangle when we know all 3 sides.

The minus sign goes with the side opposite the angle we are finding.

The Cosine Rule

bc

acbA

2cos

222 Acos

Solution: Let’s find A first

6e.g. 2 Find all the

angles in triangle ABC B

A 9

5

C

938A77780cos A)5)(9(2

659 222

We can now use the Cosine rule again or switch to the Sine rule. If we use the Sine rule, we must avoid the largest angle ( opposite the longest side ) as we don’t know whether it is less than or greater than . 90

The Cosine Rule

OR: Using the Sine rule for C :631C

6

938sin5sin

C

EITHER: Using the Cosine rule for B or C: 5109 B

)6)(5(2

965cos

222 B 3330cos B

a

A

c

C sinsin

e.g.

6

B

A 9

5

C

938A

6319385109180 C Then

5109938631180 B Then

The Cosine Rule

The Cosine Rule

SUMMARY

Abccba cos2222

• In a triangle that isn’t right angled, if we know 2 sides and the angle formed by the 2 sides, we use

• If we know 3 sides, we use

bc

acbA

2cos

222

to find the 3rd side.

to find any angle.