**Cosine rule**

Plenary

The diagram shows two yachts A and B that have sailed from the harbour at P. Yacht A has sailed for 5.8 km on a bearing of 165 from P. Yacht B has sailed for 5.2 km on a bearing of 200 from P. Work out the distance between A and B.

**L.O. - Apply the cosine rule to non-right triangles.**

Starter

Work out the missing length in the triangle.

What happens to the length if the right angle becomes an acute angle? Or an obtuse angle?

x

15 cm

8 cm

Starter

Work out the missing length in the triangle.

17 cm

What happens to the length if the right angle becomes an acute angle? Or an obtuse angle?

If the right angle becomes acute, x becomes shorter.

If the right angle becomes obtuse, x becomes longer.

x

15 cm

8 cm

Cosine Rule

For a non-right angled triangle, Pythagoras' Theorem can be adjusted by taking off twice the product of the sides adjacent to the angle and the cosine of the angle.

C

a

b

c = a + b - 2abCosC

c

2

2

2

Cosine Rule

Find the size of the third side, x, in this triangle.

123

9.5 cm

6 cm

o

x cm

Cosine Rule

Find the size of the third side, x, in this triangle.

123

9.5 cm

6 cm

o

x cm

x = 6 + 9.5 - 2(6)(9.5)Cos 123

x = 36 + 90.25 - 114Cos123

x = 188.33885

x = 188.33885 = 13.7 cm

2

2

2

2

2

Cosine Rule

Find the size of the third side, x, in this triangle.

x

9.5 cm

6 cm

o

20 cm

Cosine Rule

Find the size of the third side, x, in this triangle.

x

9.5 cm

6 cm

o

15 cm

15 = 6 + 9.5 - 2(6)(9.5)Cos x

225 = 36 + 90.25 - 114Cos x

114 Cos x = -98.75

Cox x = -0.86622807

2

2

2

x = cos (-0.86622807)

x = 150

-1

o

Activity

Complete the Cosine rule worksheet

Activity

Complete the Cosine rule worksheet

o

o

Plenary

The diagram shows two yachts A and B that have sailed from the harbour at P. Yacht A has sailed for 5.8 km on a bearing of 165 from P. Yacht B has sailed for 5.2 km on a bearing of 200 from P. Work out the distance between A and B.

o

o

AB = 5.2 + 5.8 - 2(5.2)(5.8)Cos 35

AB = 27.04 + 33.64 - 60.32Cos 35

AB = 11.26874869...

AB = 11.26874869 = 3.36 km

35

o

2

2

2

2

2

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**Worked**

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