Sine & Cosine Rules (OCR GCSE Maths)

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Sine Rule

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

fraction numerator a over denominator sin space A blank end fraction equals blank fraction numerator b over denominator sin space B end fraction equals blank fraction numerator c over denominator sin space C blank end fraction

    • 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

Non-Right-Angled Triangles Diagram 1a

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

fraction numerator sin space A blank over denominator a end fraction equals blank fraction numerator sin space B blank over denominator b end fraction equals blank fraction numerator sin space C blank over denominator c end fraction

  • 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

Examiner Tip

  • Remember to check that your calculator is in degrees mode!

Worked example

The following diagram shows triangle ABC.  AB space equals space 8.1 space cm, BC space equals space 12.3 space cm, straight B straight C with hat on top straight A equals 27 degree. The angle straight B straight A with hat on top straight C is acute.

3-3-2-sine-rule-we-question

Use the sine rule to calculate the value of:

i)
x,

 3-3-2-ai-sl-sine-rule-we-solution-i

ii)
y.

3-3-2-ai-sl-sine-rule-we-solution-ii

Cosine Rule

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

a squared equals b squared plus c squared minus 2 b c space cos space A   ;     cos space A blank equals blank fraction numerator b to the power of 2 blank end exponent plus blank c squared minus blank a squared over denominator 2 b c end fraction

    • 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 a squared space equals space b squared space plus space c squared space minus space 2 b c space cos A to find an unknown side
  • Use the formula cos space A blank equals blank fraction numerator b to the power of 2 blank end exponent plus blank c squared space minus blank a squared over denominator 2 b c end fraction  to find an unknown angle
    • A is the angle between sides b and c
  • Substitute the values you have into the formula and solve

Examiner Tip

  • Remember to check that your calculator is in degrees mode!

Worked example

The following diagram shows triangle ABC. AB space equals space 4.2 space kmBC space equals space 3.8 space km, AC space equals space 7.1 space km.

3-3-2-cosine-rule-we-question

Calculate the value of angle ABC.

4-11-1-cosine-rule-new-we-solution

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.