Sine & Cosine Rules (AQA GCSE Maths)

What is the sine rule?

• The sine rule allows us to find missing side lengths or angles in non-right-angled triangles
• It states that for any triangle with angles A, B and C

• • Where
    • a is the side opposite angle A
    • b is the side opposite angle B
    • c is the side opposite angle C
• You not need to remember it
• Sin 90° = 1 so if one of the angles is 90° this becomes SOH from SOHCAHTOA

How can we use the sine rule to find missing side lengths or angles?

• The sine rule can be used when you have any opposite pairs of sides and angles
• Always start by labelling your triangle with the angles and sides
  • Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters
• Use the formula in the formula booklet to find the length of a side
• To find a missing angle you can rearrange the formula and use the form

• Substitute the values you have into the formula and solve
  • This will always give you an acute angle
  • If you know the angle is obtuse then subtract this value from 180

What is the cosine rule?

• The cosine rule allows us to find missing side lengths or angles in non-right-angled triangles
• It states that for any triangle

   ;     

• • Where
    • a is the side opposite angle A
    • b and c are the other two sides
• You are not given either formula
  • You could memorise both of them
  • Or you could memorise the first one and rearrange it each time you use it
• Cos 90° = 0 so if A = 90° this becomes Pythagoras’ Theorem

How can we use the cosine rule to find missing side lengths or angles?

• The cosine rule can be used when you have two sides and the angle between them or all three sides
• Always start by labelling your triangle with the angles and sides
  • Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters
• Use the formula  to find an unknown side
• Use the formula  to find an unknown angle
  • A is the angle between sides b and c
• Substitute the values you have into the formula and solve