a2 = b2 + c2 – 2bcCosA

Applying the same method as earlier to the other sides produce similar formulae

for b and c. namely:b2 = a2 + c2 – 2acCosB

c2 = a2 + b2 – 2abCosC

A

B

C

a

b

c

The Cosine Rule

The Cosine rule can be used to find:

1. An unknown side when two sides of the triangle and the included angle are given.

2. An unknown angle when 3 sides are given.

Finding an unknown side.

a2 = b2 + c2 – 2bcCosAThe Cosine Rule

To find an unknown side we need 2 sides and the included angle.

a2 = 82 + 9.62 – 2 x 8 x 9.6 x Cos 40o

a = (82 + 9.62 – 2 x 8 x 9.6 x Cos 40o)

a = 6.2 cm (1 dp)

m2 = 5.42 + 7.72 – 2 x 5.4 x 7.7 x Cos 65o

m = (5.42 + 7.72 – 2 x 5.4 x 7.7 x Cos 65o)

m = 7.3 cm (1 dp)

Not to scale

8 cm

9.6 cma

1.

40o

2.7.7 cm

5.4 cm65o

m

85 m

100 m15o3.

p

p2 = 852 + 1002 – 2 x 85 x 100 x Cos 15o

p = (852 + 1002 – 2 x 85 x 100 x Cos 15o)

p = 28.4 m (1 dp)

a2 = b2 + c2 – 2bcCosAThe Cosine Rule

Application Problem

A fishing boat leaves a harbour (H) and travels due East for 40 miles to a marker buoy (B). At B the boat turns left onto a bearing of 035o and sails to a lighthouse (L) 24 miles away. It then returns to harbour.

(a) Make a sketch of the journey

(b) Find the total distance travelled by the boat. (nearest mile)

H40 miles

24 miles

B

L

125o

HL2 = 402 + 242 – 2 x 40 x 24 x Cos 1250

HL = (402 + 242 – 2 x 40 x 24 x Cos 1250)

= 57 miles

Total distance = 57 + 64 = 121 miles.

An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 430 miles North to a point P before turning left onto a bearing of 230o to a second point Q, 570 miles away. It then returns to base.

(a) Make a sketch of the flight.

(b) Find the total distance flown by the aircraft. (nearest mile)

The Cosine Rule a2 = b2 + c2 – 2bcCosA

QW2 = 4302 + 5702 – 2 x 430 x 570 x Cos 500

QW = (4302 + 5702 – 2 x 430 x 570 x Cos 500)

= 441 miles

Total distance = 1000 + 441 = 1441 miles.

50o

Not to Scale

P

570 miles

W

430 miles

Q

2 2 2 2a b c bcCosA 2 2 22bcCosA b c a

2 2 2

2b c a

CosAbc

A

B

C

a

b

c

The Cosine Rule

To find unknown angles the 3 formula for sides need to be re-arranged in terms of CosA, B or C.

a2 = b2 + c2 – 2bcCosAb2 = a2 + c2 – 2acCosBc2 = a2 + b2 – 2abCosC

2 2 2

2a c b

CosBac

2 2 2

2a b c

CosCab

Similarly

Not to scale

8 cm

9.6 cm6.2

1.

A

2.7.7 cm

5.4 cmP

7.3 cm

85 m

100 m3. R

28.4 m

The Cosine Rule

To find an unknown angle we need 3 given sides.

2 2 2

2b c a

CosAbc

2 2 28 9.6 6.22 8 9.6

CosAx x

2 2 25.4 7.7 7.32 5.4 7.7

CosPx x

2 2 2100 85 28.42 100 85

CosRx x

A 40o 65oP

15oR

A fishing boat leaves a harbour (H) and travels due East for 40 miles to a marker buoy (B). At B the boat turns left and sails for 24 miles to a lighthouse (L). It then returns to harbour, a distance of 57 miles.

(a) Make a sketch of the journey.

(b) Find the bearing of the lighthouse from the harbour. (nearest degree)

The Cosine Rule

Application Problems

2 2 2

2b c a

CosAbc

H40 miles

24 miles

B

L

57 miles

A

2 2 257 40 242 57 40

CosAx x

A 20.4o

90 0 020.4 7 oBearing

2 2 2

2b c a

CosAbc

The Cosine Rule a2 = b2 + c2 – 2bcCosA

An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 530 miles North to a point (P) as shown, It then turns left and flies to a point (Q), 670 miles away. Finally it flies back to base, a distance of 520 miles.

Find the bearing of Q from point P.

2 2 2530 670 5202 530 670

CosPx x

48.7oP

180 22948.7 oBearing

P

670 miles

W

530 miles

Not to Scale

Q

520 miles