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Deciding the Trig Rule (Cambridge (CIE) IGCSE International Maths)

Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 1Cambridge (CIE) Maths: ExtendedEdexcel Maths A: Higher

Applications of Trigonometry

How do I decide which trig rule to use?

  • Different rules are required depending on the question

    • You need to be able to decide which is appropriate to use

    • Think about what information you have and what you want to find

  • This table summarises the possibilities:

If you know

And you want to know

Use

Two sides and an angle opposite one of the sides

The angle opposite the other side

Sine rule

Two angles and a side opposite one of the angles

The side opposite the other angle

Sine rule

Two sides and the angle between them

The third side

Cosine rule

All three sides

Any angle

Cosine rule

Two sides and the angle between them

The area of the triangle

Area of a triangle rule

Flow chart to determine which rule or formula to use.

Can I use multiple trig rules in the same question?

  • Harder questions will require you to use more than one trig rule

    • For example, you may need the sine rule followed by the cosine rule

  • The area formula only works for an angle between two sides

    • If you are not given this setup, you may need to use the sine or cosine rule first

  • If it looks like no rule would work, remember that all angles in a triangle sum to 180

    • This often helps to find a missing angle

Examiner Tips and Tricks

  • Look at the number of marks for a question - if it is a lot, you are likely to need more than one trig rule!

Worked Example

Find the area of the triangle below.

General-Triangle-with-values-2, IGCSE & GCSE Maths revision notes

The area of a triangle can be found using the formula A equals 1 half a b space sin space C

The three side lengths are known , but we need to find an angle in order to calculate the area
Because we know all three sides, any of the angles could be found

Find angle ABC using the cosine rule

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

table row cell A C squared end cell equals cell A B squared plus B C squared minus 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 squared end cell equals cell 7.4 squared plus 4.8 squared minus 2 open parentheses 7.4 close parentheses open parentheses 4.8 close parentheses space cos space A B C end cell end table

Rearrange to makespace cos space A B C the subject

table row cell 4.4 squared minus open parentheses space 7.4 squared plus 4.8 squared close parentheses end cell equals cell negative 2 open parentheses 7.4 close parentheses open parentheses 4.8 close parentheses space cos space A B C end cell row cell cos space A B C end cell equals cell fraction numerator 4.4 squared minus 7.4 squared minus 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 end cell equals cell 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 end cell equals cell cos to the power of negative 1 end exponent open parentheses 487 over 592 close parentheses end cell row blank equals cell 34.65054... end cell end table

Now we can find the area of the triangle using the formula and angle ABC as the known angle

table row Area equals cell 1 half cross times 4.8 cross times 7.4 cross times sin space 34.65054... end cell row blank equals cell 10.09779... end cell end table

bold Area bold equals bold 10 bold. bold 1 bold space bold cm to the power of bold 2 bold space stretchy left parenthesis 3 space s. f. stretchy right parenthesis

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    Author
    Reviewer
    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.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.