Joan Ridgway

If the question concerns lengths or angles in a triangle, you may need the sine rule or the cosine rule.

First, decide if the triangle is right-angled.

Then, decide whether an angle is involved at all.

If it is a right-angled triangle, and there are angles involved, you will need straightforward Trigonometry, using Sin, Cos and Tan.

If the triangle is not right-angled, you may need the Sine Rule or the Cosine Rule

If it is a right-angled triangle, and there are no angles involved, you will need Pythagoras’ Theorem

In any triangle ABC

The Sine Rule:

A B

C

ab

c

C

c

B

b

A

a

sinsinsin

or

c

C

b

B

a

A sinsinsin

Not right-angled!

You do not have to learn the Sine Rule or the Cosine Rule!

They are always given to you at the front of the Exam Paper.

You just have to know when and how to use them!

The Sine Rule:

A B

C

ab

c

You can only use the Sine Rule if you have a “matching pair”.

You have to know one angle, and the side opposite it.

The Sine Rule:

A B

C

ab

c

You can only use the Sine Rule if you have a “matching pair”.

You have to know one angle, and the side opposite it.

Then if you have just one other side or angle, you can use the Sine Rule to find any of the other angles or sides.

10cm

x

65°

Finding the missing side:

Is it a right-angled triangle?

Is there a matching pair?

No

Yes

40°

Not to scale

10cm

65°

Finding the missing side:

Is it a right-angled triangle?

Is there a matching pair?

No

Yes

Label the sides and angles.

A

B

C

ab

c

40°x

Use the Sine Rule

Not to scale

10cm

65°

Finding the missing side:

A

B

C

ab

c

40°x

C

c

B

b

A

a

sinsinsin

We don’t need the “C” bit of the formula.

Because we are trying to find a missing length of a side, the little letters are on top

Not to scale

10cm

65°

Finding the missing side:

A

B

C

ab

c

40°x

B

b

A

a

sinsin

Fill in the bits you know.

Because we are trying to find a missing length of a side, the little letters are on top

Not to scale

10cm

65°

Finding the missing side:

A

B

C

ab

c

40°x

B

b

A

a

sinsin

Fill in the bits you know.

65sin

10

40sin

x

Not to scale

10cm

65°

Finding the missing side:

A

B

C

ab

c

40°x

B

b

A

a

sinsin

65sin

10

40sin

x

40sin65sin

10x

09.7x cm

Not to scale

10cm

7.1cm

65°

Finding the missing angle:

Is it a right-angled triangle?

Is there a matching pair?

No

Yes

θ°

Not to scale

Finding the missing angle:

Is it a right-angled triangle?

Is there a matching pair?

No

Yes

Label the sides and angles.

A

B

C

ab

c

Use the Sine Rule

10cm

7.1cm

65°

θ°

Not to scale

Finding the missing angle:

We don’t need the “C” bit of the formula.

A

B

C

ab

c

10cm

7.1cm

65°

θ°

Because we are trying to find a missing angle, the formula is the other way up. c

C

b

B

a

A sinsinsin

Not to scale

Finding the missing angle:

Fill in the bits you know.

Because we are trying to find a missing angle, the formula is the other way up.

A

B

C

ab

c

10cm

7.1cm

65°

θ°

b

B

a

A sinsin

Not to scale

Finding the missing angle:

Fill in the bits you know.

10

65sin

1.7

sin

A

B

C

ab

c

10cm

7.1cm

65°

θ°

b

B

a

A sinsin

Not to scale

Finding the missing angle:

1.710

65sinsin

.....6434785.0sin

A

B

C

ab

c

10cm

7.1cm

65°

θ°

10

65sin

1.7

sin

05.40Shift Sin =

b

B

a

A sinsin

Not to scale

If the triangle is not right-angled, and there is not a matching pair, you will need the Cosine Rule.

The Cosine Rule:

A B

C

ab

c

In any triangle ABC Abccba cos2222

Finding the missing side:

Is it a right-angled triangle?

Is there a matching pair?

No

No

9cm

12cm

20°A

C

B

x

Use the Cosine Rule

Label the sides and angles, calling the given angle “A” and the missing side “a”.

ab

c Not to scale

Finding the missing side:

9cm

12cm

20°A

C

B

x ab

c

Fill in the bits you know.

Abccba cos2222

x = 4.69cm

20cos9122912 222a

)20cos9122(912 222 a

........026.22a69.4a

Not to scale

.......026.222 a

Finding the missing side:

Is it a right-angled triangle?

Is there a matching pair?

No

No

8km

5km

130°

A man starts at the village of Chartham and walks 5 km due South to Aylesham. Then he walks another 8 km on a bearing of 130° to Barham.

What is the direct distance between Chartham and Barham, in a straight line?

A

C

B

First, draw a sketch.

Use the Cosine Rule

Not to scale

Finding the missing side:

a

8km

5km

130°

A man starts at the village of Chartham and walks 5 km due South to Aylesham. Then he walks another 8 km to on a bearing of 130° to Barham.

What is the direct distance between Chartham and Barham, in a straight line?

A

C

B

Abccba cos2222 Call the missing length you want to find “a”Label the other sides

b

ca² = 5² + 8² - 2 x 5 x 8 x cos130°

a² = 25 + 64 - 80cos130°

a² = 140.42

a = 11.85 11.85km

Not to scale

Is it a right-angled triangle?

Is there a matching pair?

No

No

Use the Cosine Rule

a9cm6cm

A

C

B

b

c10cmθ°

Label the sides and angles, calling the missing angle “A”

Finding the missing angle θ:

Not to scale

Finding the missing angle θ:

a9cm6cm

A

C

B

b

c10cmθ°

Abccba cos2222

Not to scale

This can be rearranged to:

bc

acbA

2cos

222

1062

9106cos

222

A

458333333.0cos A

Shift Cos =

7.62)458333333.0(cos 1

72.62

The diagram shows the route taken of an orienteering competition.The shape of the course is a quadrilateral ABCD.AB = 4 km, BC = 4.5 km and CD = 5 km.The angle at B is 70° and the angle at D is 52°

Calculate the angle CAD.

4km 4.5km

A

5kmNot to scale

B

C

D

70°

52°

θ°x

We haven’t got enough information

about the triangle ACD to find

First find x, by looking at just the upper triangle, because this will give us enough information in the lower triangle to find .

4km 4.5km

A

B

C

70°

x

Is it a right-angled triangle?

Is there a matching pair?

No

No

Use the Cosine Rule

Label the sides and angles, calling the given angle “A” and the missing side “a”.

A

a

B

b

C

c

4km 4.5km

A

B

C

70°

x

A

a

B

b

C

c

Abccba cos2222 70cos45.4245.4 222 a

93727484.232 a

93727484.23a

892573437.4ax = 4.893km (to 3 d.p.)

4km 4.5km

A

5kmNot to scale

B

C

D

70°

52°

θ°4.893km

Now look at the lower triangle

A

5km

C

D

52°

θ°4.893km

Now look at the lower triangle

Is it a right-angled triangle?

Is there a matching pair?

No

Yes

Label the sides and angles.

Use the Sine Rule

B

b

a

A

A

5km

C

D

52°

θ°4.893km

B

b

b

B

a

A sinsin

893.4

52sin

5

sin

5893.4

52sinsin

805242952.0sin

6.53Shift Sin =

A

a

The diagram shows the route taken of an orienteering competition.The shape of the course is a quadrilateral ABCD.AB = 4 km, BC = 4.5 km and CD = 5 km.The angle at B is 70° and the angle at D is 52°

Calculate the angle CAD.

4km 4.5km

A

5kmNot to scale

B

D

70°

52°

θ°

6.53

Angle CAD = 53.6°