O LevelMathsCambridgeRevision NotesPythagoras & TrigonometrySine, Cosine Rule & Area of TrianglesApplications of Trigonometry

Applications of Trigonometry (Cambridge O Level 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

  • Make sure you can do this rearrangement, or remember this form of the formula

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 other angle

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 notes

a)
Find the size of angle A B C.
  

The three side lengths are known and we want to find an angle so use the cosine rule. 

Cosine Rule:  c to the power of 2 space end exponent equals space a to the power of 2 space end exponent plus space b to the power of 2 space end exponent minus space 2 a b space cos space C, where C is the angle opposite side c.
  

table row cell A C to the power of 2 space end exponent end cell equals cell space A B to the power of 2 space end exponent plus space B C to the power of 2 space end exponent minus space 2 open parentheses A B close parentheses open parentheses B C close parentheses space cos space A B C end cell row cell 4.4 to the power of 2 space end exponent end cell equals cell space 7.4 squared space plus space 4.8 to the power of 2 space end exponent minus space 2 open parentheses 7.4 close parentheses open parentheses 4.8 close parentheses cos space A B C end cell end table

Rearrange to makespace cos A B C the subject.

table row cell 4.4 to the power of 2 space end exponent minus open parentheses space 7.4 squared space plus space 4.8 to the power of 2 space end exponent close parentheses end cell equals cell negative space 2 open parentheses 7.4 close parentheses open parentheses 4.8 close parentheses cos space A B C end cell row cell cos space A B C space end cell equals cell space fraction numerator 4.4 to the power of 2 space end exponent minus space 7.4 squared space space minus space 4.8 to the power of 2 space end exponent over denominator negative 2 open parentheses 7.4 close parentheses open parentheses 4.8 close parentheses end fraction end cell end table

Use your calculator to find the value of cos space A B C.

table row cell cos space A B C space end cell equals cell space 487 over 592 end cell end table

Use the cos-1 button on your calculator to find the value of A B C.

table row cell A B C space end cell equals cell space cos to the power of negative 1 end exponent space open parentheses 487 over 592 close parentheses end cell row blank equals cell space 34.650542... end cell end table

b)
Given that angle A C B is obtuse, use the Sine Rule and your answer from (a) to find the size of angle A C B.
Give your answers accurate to 1 d.p.
   

Substitute the values of  A B space equals space 7.4 comma space space A C space equals space 4.4 space space and space space A B C space equals space 34.650542... degree into the sine rule. 

Sine Rule:  fraction numerator space a over denominator sin space A end fraction equals space fraction numerator space b over denominator sin space B end fraction space equals space fraction numerator space c over denominator sin space C end fraction.

fraction numerator 7.4 over denominator space sin space A C B end fraction equals space fraction numerator space 4.4 over denominator sin space 34.650542... degree end fraction

Rearrange the equation to find the value of A C B.

table row cell sin A C B space end cell equals cell space fraction numerator space 7.4 space sin space 34.650542... degree over denominator 4.4 end fraction end cell row blank equals cell space 0.9562307... end cell row cell A C B space end cell equals cell space sin to the power of negative 1 end exponent open parentheses 0.9562307... close parentheses space equals space 72.985502... end cell end table

  

Subtract from 180° to find the value of obtuse angle A C B.

table row cell A C B space end cell equals cell space 180 space minus space 72.985502... space equals space 107.014497... end cell end table

  

bold italic A bold italic C bold italic B bold space bold equals bold space bold 107 bold. bold 0 bold degree bold space stretchy left parenthesis 1 space d. p. stretchy right parenthesis

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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.