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Trigonometric Identities (AQA GCSE Further Maths)

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Trigonometric Identities

What is a trigonometric identity?

  • Trigonometric identities (trig identity) are statements that are true for all values (of x or theta)

    • They are used to help simplify trigonometric equations before solving them

    • Sometimes you may see identities written with the symbol ≡

      • This means 'identical to'

What trigonometric identities do I need to know?

  • There are two trig identities you need to know and use

    • bold tan bold space bold italic theta bold identical to fraction numerator bold sin bold space bold theta over denominator bold cos bold space bold theta end fraction

      • This is the identity for tan θ

    • bold sin to the power of bold 2 space bold italic theta bold plus bold cos to the power of bold 2 bold space bold italic theta bold identical to bold 1

      • You may see this referred to as the Pythagorean identity

      • Note that the notation sin space squared theta is the same as left parenthesis sin space theta right parenthesis to the power of space 2 end exponent

      • Similar for cos squared space theta

  • Both identities are given on the formulae sheet

Where do the trigonometric identities come from?

  • You do not need to know the proof for these identities

    • However it is a good idea to know where they come from

  • The identity for tan space theta can be seen by diving sin space theta by cos space theta using their definitions from SOHCAHTOA

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

  • The Pythagorean identity can be seen by considering a right-triangle with a hypotenuse of 1

    • Then using Pythagoras' theorem (a squared equals b squared plus c squared, where a is the hypotenuse)

      • 1 squared equals O squared plus A squared

      • From SOHCAHTOA, table row cell sin space theta end cell equals cell O over H equals O over 1 comma space space therefore O equals sin space theta end cell end table and cos space theta equals A over H equals A over 1 comma space therefore A equals cos space theta

      • And so sin squared space theta plus cos squared space theta blank equals blank 1

How are the trigonometric identities used?

  • Most commonly trig identities are used to rewrite an equation

  • Rearrangements of the Pythagorean identity are very useful for rewriting equations

    • This allows us to write equations in terms of sine or cosine only (making them easier to solve)

      • sin to the power of 2 space end exponent theta equals blank 1 minus space cos to the power of 2 space end exponent theta

      • cos to the power of 2 space end exponent theta equals blank 1 minus space sin to the power of 2 space end exponent theta

Examiner Tips and Tricks

  • If you are asked to show that one expression is identical (≡) to another, look for anything that has gone missing!

    • e.g.  if tan space x is in the original expression but not the 'answer' it must have been replaced by fraction numerator sin space x over denominator cos space x end fraction

Worked Example

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

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