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Trigonometric Identities (Cambridge O Level Additional Maths)

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Amber

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Simple Trig 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
      • This formula does not appear in the list of formulae
    • 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
      • This formula appears in the list of formulae
  • 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 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

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

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.

Substitute sin squared x equals 1 minus cos squared x into the equation to form a quadratic in terms of cos x.

2 open parentheses 1 minus cos squared x close parentheses minus cos x equals 0

Expand the bracket.

2 minus 2 cos squared x minus cos x equals 0

Rewrite in the required form.

bold 2 bold cos to the power of bold 2 bold italic x bold plus bold cos bold italic x bold minus bold 2 bold equals bold 0

or

bold minus bold 2 bold cos to the power of bold 2 bold italic x bold minus bold cos bold italic x bold plus bold 2 bold equals bold 0

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Further Trig Identities

What are the identities linking tan, sec, cot, and cosec?

  • Aside from the Pythagorean identity sin2x + cos2x = 1 there are two further Pythagorean identities you will need to learn
    •  1 plus tan squared space invisible function application theta equals sec squared space invisible function application theta
    •  1 plus cot squared space invisible function application theta equals cosec squared space invisible function application theta
    • Both can be found in the list of formulae
  • Both of these identities can be derived from sin2x + cos2x = 1 
    • To derive the identity for sec2x divide sin2x + cos2x = 1 by cos2x
    • To derive the identity for cosec2x divide sin2x + cos2x = 1 by sin2x

Deriving the Pythagorean identities for the reciprocal trig functions

How do I prove new trigonometric identities?

  • You can use trigonometric identities you already know to prove new identities
  • To prove an identity start on one side and proceed step by step until you get to the other side
    • It is more common to start on the left hand side but you can start a proof from either end
    • Occasionally it is easier to show that one side subtracted from the other is zero
    • You should not work on both sides simultaneously
  • Look for anything that could be a part of one of the above identities on either side
    • For example if you see sin squared theta you can replace it with 1 space minus space cos squared theta
  • Look for ways of reducing the number of different trigonometric functions there are within the identity
    • For example if the identity contains tan θ, cot θ and cosec θ you could try 
      • Using the identities tan θ = 1/cot θ and 1 + cot2 θ = cosec2 θ to write it all in terms of cot θ
      • Or rewriting it all in terms of sin θ and cos θ and simplifying
  • Often you may need to trial a few different methods before finding the correct one
  • Always keep an eye on the 'target' expression – this can help suggest what identities to use

Examiner Tip

  • Writing down all the identities you know can be a good way to spot how to get started on a question

Worked example

Solve the equation 9 sec2 θ – 11 = 3 tan θ in the interval 0 ≤ θ ≤ 2π. Give your answers to three decimal places.

The squared term can be rewritten using the identity sec squared theta equals 1 plus tan squared theta. Substitute this into the equation.

9 open parentheses 1 plus tan squared theta close parentheses minus 11 equals 3 tan theta

Rearrange to form a quadratic.

9 plus 9 tan squared theta minus 11 equals 3 tan theta 9 tan squared theta minus 3 tan theta minus 2 equals 0

Treat as a quadratic using x equals tan theta.

9 x squared minus 3 x minus 2 equals 0 open parentheses 3 x minus 2 close parentheses open parentheses 3 x plus 1 close parentheses equals 0 x equals 2 over 3 comma space x equals negative 1 third

Substitute x equals tan theta back in and find the principal value for each equation.

tan theta equals 2 over 3 rightwards arrow tan to the power of negative 1 end exponent open parentheses 2 over 3 close parentheses equals 0.5880... tan theta equals negative 1 third rightwards arrow tan to the power of negative 1 end exponent open parentheses negative 1 third close parentheses equals negative 0.3217...

Find the other solutions in the interval by adding multiples of π.

straight pi plus 0.5880... equals 3.7295... straight pi plus open parentheses negative 0.3217... close parentheses equals 2.8198... 2 straight pi plus open parentheses negative 0.3217... close parentheses equals 5.9614...

Ignore the solution that is outside the interval (-0.3217...). Round the other four to three decimal places.

bold italic theta bold equals bold 0 bold. bold 588 bold comma bold space bold 2 bold. bold 820 bold comma bold space bold 3 bold. bold 730 bold comma bold space bold 5 bold. bold 961

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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.