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GCSEMathsOCRRevision Notes10. MensurationSine, Cosine Rule & Area of TrianglesApplications of Trigonometry

Applications of Trigonometry (OCR GCSE Maths)

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Applications of Trigonometry

Choosing which rule or formula to use

  • It is important to be able to decide which Rule or Formula to use to answer a question
  • This table summarises the possibilities:

Sine & Cosine Rules, Area of Triangle – Harder table, IGCSE & GCSE Maths revision notes

Non-Right-Angled Triangles Diagram 2

Using the cosine rules to find angles

  • The Cosine Rule can be rearranged to give: 

cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction

  • When using the inverse cosine function (i.e. cos to the power of negative 1 end exponent) we can use this to find the size of angle A:

A equals cos to the power of negative 1 end exponent open parentheses fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction close parentheses

  • This form of the formula is not on your exam formula sheet, so make sure you can do the rearrangement yourself!

Using the sine rule to find angles

Sine-Rule-Ambiguous-case, IGCSE & GCSE Maths revision notes

  • If all we know are the lengths of A B and B C and the size of angle B A C, there are two possible triangles that could be drawn
    • one with side B C subscript 1 (and angle x equals 102.8 degree)
    • the other with side B C subscript 2 (and angle y equals 77.2 degree)
    • Using your calculator and the Sine Rule would only find you the possibility with angle y
    • You may need to subtract your answer from 180° to find the angle you need 

Examiner Tip

  • In more involved exam questions, you may have to use both the Cosine Rule and the Sine Rule over several steps to find the final answer
  • If your calculator gives you a ‘Maths ERROR’ message when trying to find an angle using the Cosine Rule, you probably subtracted things the wrong way around when you rearranged the formula
  • The Sine Rule can also be written ‘flipped over’:

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

    • This is more useful when we are using the rule to find angles
    • When finding angles with the Sine Rule, use the info in the question to decide whether you have the acute angle case (ie the calculator value) or the obtuse angle case (ie, minus the calculator value)
  • The Cosine Rule will never give you an ambiguous answer for an angle – as long as you put the right things into the calculator, the answer that comes out will be the correct angle

Worked example

In the following triangle:General-Triangle-with-values-2, IGCSE & GCSE Maths revision notesa) Find the size of angle ABC.

b) Given that angle ACB is obtuse, use the Sine Rule and your answer from (a) to find the size of angle ABC.

Give your answers accurate to 1 d.p.

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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.