Higher only · sine and cosine rules

Sine rule or
cosine rule?
How to choose.

it starts with how
you label the triangle

when there is no right angle —

Any triangle

SOH CAH TOA needs 90°

these two rules do not

step 1 of 4

Label the triangle

1. Capital letters for the angles: A, B, C.

2. Small letters for the sides: a, b, c.

✓ Side a is always opposite angle A

step 2 of 4

The sine rule

1. You know a pair: 75° and its side, 10 cm.

2. Write each side over the sin
of its opposite angle.

3. a = 10 × sin 40° ÷ sin 75°

✓ a = 6.65 cm (3 s.f.)

step 3 of 4

The cosine rule

1. Two sides and the angle between them.

2. a² = 8² + 5² − 2 × 8 × 5 × cos 60°

3. a² = 89 − 40 = 49

✓ a = √49 = 7 cm

step 4 of 4

Which rule?

1. An angle and its opposite side?
Sine rule.

2. Two sides and the angle between,
or all three sides? Cosine rule.

✓ A right angle? Back to SOH CAH TOA

the two rules —

Sine rule

asin A = bsin B = csin C

Cosine rule

a² = b² + c² − 2bc cos A

Finding an angle with the sine
rule? Flip it: sin A over a.

The
Maths Habit

Download The Maths Habit app
to test your knowledge