Sine & Cosine Rules (CIE IGCSE Maths: Extended)

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
    •  is the side opposite angle A
    •  is the side opposite angle B
    •  is the side opposite angle C
• 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 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

What is the ambiguous case of the sine rule?

• If the sine rule is used in a triangle given two sides and an angle which is not the angle between them there may be more than one possible triangle which could be drawn
• The side opposite the given angle could be in two possible positions
• This will create two possible values for each of the missing angles and two possible lengths for the missing side
• The two angles found opposite the given side (not the ambiguous side) will add up to 180°
  • The question will usually tell you whether the angle you are looking for is acute or obtuse
  • The sine rule will always give you the acute option but you can subtract from 180° to find the obtuse angle
  • Sometimes the obtuse angle will not be valid as it could cause the sum of the three interior angles of the triangle to exceed 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
• Both of these formulae are rearrangements of the same thing
  • The first version is used to find a missing side
  • The second version is a rearrangement of this and can be used to find a missing angle
• 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