Simple Identities (DP IB Maths: AI HL)

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Simple Identities

What is a trigonometric identity?

  • Trigonometric identities 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?

  • The two trigonometric identities you must know are
    • tan space theta space equals space fraction numerator sin space theta over denominator cos space theta end fraction
      • This is the identity for tan θ
    • sin squared theta space plus space cos squared theta space equals space 1
      • This is 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
  • Both identities can be found in the formula booklet
  • Rearranging the second identity often makes it easier to work with
    • sin squared theta equals blank 1 minus space cos to the power of 2 space end exponent theta
    • cos squared theta equals blank 1 minus space sin squared theta

Where do the trigonometric identities come from?

  • You do not need to know the proof for these identities but it is a good idea to know where they come from
  • From SOHCAHTOA we know that
    • sin space theta blank equals opposite over hypotenuse equals O over H
    • cos space theta blank equals adjacent over hypotenuse equals A over H
    • tan space theta blank equals opposite over adjacent equals O over A
  • The identity for tan space theta can be seen by diving sin space theta by cos space theta
    • 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
    • This can also be seen from the unit circle by considering a right-triangle with a hypotenuse of 1
      • tan space theta space equals space O over A space equals space fraction numerator sin space theta over denominator cos space theta end fraction
  • The Pythagorean identity can be seen by considering a right-triangle on the unit circle with a hypotenuse of 1
    • Then (opposite)2 + (adjacent)2 = 1
    • Therefore sin squared space theta plus cos squared space theta blank equals blank 1
  • Considering the equation of the unit circle also shows the Pythagorean identity
    • The equation of the unit circle is space x squared space plus space y squared space equals space 1
    • The coordinates on the unit circle are left parenthesis cos space theta comma blank sin space theta right parenthesis
    • Therefore the equation of the unit circle could be written cos squared space theta plus sin squared space theta equals 1
  • A third very useful identity is sin space theta equals cos space left parenthesis 90 degree minus blank theta right parenthesis blankor sin space theta equals cos space left parenthesis pi over 2 minus blank theta right parenthesis
    • This is not included in the formula booklet but is useful to remember

How are the trigonometric identities used?

  • Most commonly trigonometric identities are used to change an equation into a form that allows it to be solved
  • They can also be used to prove further identities such as the double angle formulae

Examiner Tip

  • If you are asked to show that one thing is identical (≡) to another, look at what parts are missing –  for example, if tan x has gone it must have been substituted

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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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.