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Trigonometric Identities (Edexcel IGCSE Further Pure Maths)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Trigonometric Identities

What is a trigonometric identity?

  • Trigonometric identities are statements about trigonometric functions like sin theta, cos theta and tan theta

    • They are true for all values of the angle theta

    • They can be used to help simplify trigonometric equations before solving them

  • Sometimes you may see identities written with the symbol  identical to  instead of an equals sign

    • This means 'identical to' or 'equivalent to'

What trigonometric identities do I need to know?

  • You must know these two trigonometric identities:

    • tan theta space equals space fraction numerator sin theta over denominator cos theta end fraction

      • This is the identity for tan theta

    • sin squared theta space plus space cos squared theta space equals space 1

      • This is sometimes called the Pythagorean identity

      • Note that the notation sin squared theta is the same thing as left parenthesis sin theta right parenthesis squared

  • The second identity is often used in one of its rearranged forms

    • sin squared theta equals blank 1 minus space cos squared theta

    • cos squared theta equals blank 1 minus space sin squared theta

Examiner Tips and Tricks

  • When asked to show that one thing is equal or identical to another, look at what parts are 'missing'

    • This can help you spot which identity to use

Worked Example

Show that the equation 2 sin squared theta minus cos theta equals 0 can be written in the form  a cos squared theta plus b cos theta plus c equals 0, where a, b and c are integers to be found with a greater than 0.

Note that in the 'target' form there is no sin theta or sin squared theta

That means we want to use a substitution to get rid of the sin squared theta in the original form

We can do this using the identity sin squared theta space plus space cos squared theta space equals space 1, rearranged as  sin squared theta space equals space 1 space minus space cos squared theta

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 sin squared theta minus cos theta end cell equals cell 2 open parentheses 1 space minus space cos squared theta close parentheses minus cos theta end cell row blank equals cell 2 minus 2 cos squared theta minus cos theta end cell row blank equals cell negative 2 cos squared theta minus cos theta plus 2 end cell end table

Substitute that back into the original equation

table row cell negative 2 cos squared theta minus cos theta plus 2 end cell equals 0 end table

Multiply both sides of the equation by negative 1 to make the cos squared theta coefficient positive

This gets the equation into the required form with  a equals 2, b equals 1 and c equals negative 2

bold 2 bold cos to the power of bold 2 bold theta bold plus bold cos bold theta bold minus bold 2 bold equals bold 0

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    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

    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.