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

a b

c

Cc

Bb

Aa

sinsinsin==

or cC

bB

aA 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

a b

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

a b

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

a b

c

40° x

Use the Sine Rule

Not to scale

10cm

65°

Finding the missing side:

A

B

C

a b

c

40° x

Cc

Bb

Aa

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

a b

c

40° x

Bb

Aa

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

a b

c

40° x

Bb

Aa

sinsin=

Fill in the bits you know.

°=

° 65sin10

40sinx

Not to scale

10cm

65°

Finding the missing side:

A

B

C

a b

c

40° x

Bb

Aa

sinsin=

°=

° 65sin10

40sinx

°×°

= 40sin65sin10x

09.7=x 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

a b

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

a b

c

10cm 7.1cm

65°

θ°

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

CbB

aA 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

a b

c

10cm 7.1cm

65°

θ°

bB

aA sinsin=

Not to scale

Finding the missing angle:

Fill in the bits you know.

1065sin

1.7sin °

=θ

A

B

C

a b

c

10cm 7.1cm

65°

θ°

bB

aA sinsin=

Not to scale

Finding the missing angle:

1.71065sinsin ×°

=θ

.....6434785.0sin =θ

A

B

C

a b

c

10cm 7.1cm

72°

θ°

1065sin

1.7sin °

=θ

°= 05.40θShift Sin =

bB

aA sinsin=

Not to scale

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

The Cosine Rule:

A B

C

a b

c

In any triangle ABC Abccba cos2222 −+=

Finding the missing side:

Is it a right-angled triangle? Is there a matching pair?

No No

9km

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”.

a b

c Not to scale

Finding the missing side:

9km

12cm

20° A

C

B

x a b

c

Fill in the bits you know.

Abccba cos2222 −+=

x = 4.69cm

°×××−+= 20cos9122912 222a)20cos9122(912 222 °×××−+=a

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

Not to scale

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

c

a² = 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

a 9cm 6cm

A

C

B

b

c 10cm θ°

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

Finding the missing angle θ:

Not to scale

Finding the missing angle θ:

a 9cm 6cm

A

C

B

b

c 10cm θ°

Abccba cos2222 −+=

θcos12013681 ×−=θcos1201003681 ×−+=

θcos10621069 222 ×××−+=

12081136cos −

=θ

....4583333.0cos =θ

Shift Cos = °= 72.62θ

Not to scale