National 5MathsSQARevision NotesTrigonometric SkillsTrigonometric Equations & IdentitiesTrigonometric Identities

Trigonometric Identities (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Using trigonometric identities

What is an identity?

  • An identity can be thought of as a 'stronger form of an equation'

    • An equation like 3 x plus 4 equals 13 is only true for a particular value of x

    • But an identity like cos squared x degree plus sin squared x degree equals 1 is always true, for any values of x

      • The expressions on the two sides of an identity are 'the same' (mathematically identical)

What trigonometric identities do I need to know?

  • You need to know and be able to use the following two trigonometric identities:

    • cos squared x degree plus sin squared x degree equals 1

    • tan x degree equals fraction numerator sin x degree over denominator cos x degree end fraction

How do I use trigonometric identities?

  • A question may ask you to simplify a trigonometric expression, or to rewrite it in a different form

  • You may need to use substitution

    • For example replacing tan x degree with fraction numerator sin x degree over denominator cos x degree end fraction or vice versa

    • space sin x degree cos x degree tan x degree space equals space sin x degree up diagonal strike cos x degree end strike open parentheses fraction numerator sin x degree over denominator up diagonal strike cos x degree end strike end fraction close parentheses space equals sin squared x degree

  • You may need to use algebraic 'tricks' like factorising to allow you to use an identity

    • 3 sin squared x degree plus 3 cos squared x degree equals 3 open parentheses sin squared x degree plus cos squared x degree close parentheses equals 3 open parentheses 1 close parentheses equals 3

  • In general

    • Look out for places where one side of an identity (or something close to it) appears in a question

      • You may need to replace this with the other side of the identity

      • Some algebra may be needed first to get an exact match

    • Keep an eye on the target form that you are trying to get an expression into

      • This may help you decide which identity needs to be used

Examiner Tips and Tricks

The identity cos squared x degree plus sin squared x degree equals 1 can be rewritten in the following two forms:

  • cos squared x degree equals 1 minus sin squared x degree

  • sin squared x degree equals 1 minus cos squared x degree

This can be used to rewrite an expression in cos squared x degree as an expression in sin squared x degree, or vice versa.

Worked Example

Express fraction numerator sin x degree cos x degree over denominator tan x degree end fraction in its simplest form.

Show your working.

Answer:

Substitute fraction numerator sin x degree over denominator cos x degree end fraction in place of tan x degree

fraction numerator sin x degree cos x degree over denominator tan x degree end fraction equals fraction numerator sin x degree cos x degree over denominator open parentheses fraction numerator sin x degree over denominator cos x degree end fraction close parentheses end fraction

Rewrite as a division and use the rules for dividing by fractions

equals sin x degree cos x degree divided by fraction numerator sin x degree over denominator cos x degree end fraction equals sin x degree cos x degree cross times fraction numerator cos x degree over denominator sin x degree end fraction

Carry out the multiplication

  • Start by cancelling common factors

equals space up diagonal strike sin x degree end strike cos x degree cross times fraction numerator cos x degree over denominator up diagonal strike sin x degree end strike end fraction equals cos squared x degree

fraction numerator sin x degree cos x degree over denominator tan x degree end fraction equals cos squared x degree

Worked Example

Express 5 minus 2 sin squared x degree in the form a plus b cos squared x degree.

Show your working.

Answer:

You are trying to rewrite an expression in sin squared x degree as an expression in cos squared x degree

  • So use space cos squared x degree plus sin squared x degree equals 1, rewritten as space sin squared x degree equals 1 minus cos squared x degree

5 minus 2 sin squared x degree equals 5 minus 2 open parentheses 1 minus cos squared x degree close parentheses

Expand the brackets and simplify

  • Be careful with the minus signs inside and outside of the brackets

equals 5 minus 2 plus 2 cos squared x degree equals 3 plus 2 cos squared x degree

That is the form you are looking for, with a equals 3 and b equals 2

5 minus 2 sin squared x degree equals 3 plus 2 cos squared x degree

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.