Trigonometric Proof (DP IB Analysis & Approaches (AA)) : Revision Note

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Trigonometric Proof

How do I prove new trigonometric identities?

  • You can use trigonometric identities you already know to prove new identities

  • Make sure you know how to find all of the trig identities in the formula booklet

    • The identity for tan, simple Pythagorean identity and the double angle identities for in and cos are in the SL section

      • tan invisible function application theta equals fraction numerator sin invisible function application theta over denominator cos invisible function application theta end fraction

      • cos squared invisible function application theta blank plus sin squared invisible function application theta equals 1

      • sin invisible function application 2 theta equals 2 sin invisible function application theta cos invisible function application theta blank

      • cos invisible function application 2 theta equals cos squared invisible function application theta minus sin squared invisible function application theta equals 2 cos squared invisible function application theta minus 1 equals 1 minus 2 sin squared invisible function application theta blank

    • The reciprocal trigonometric identities for sec and cosec, further Pythagorean identities, compound angle identities and the double angle formula for tan

      • sec invisible function application theta equals fraction numerator 1 over denominator cos invisible function application theta blank end fraction

      • cosec blank theta equals fraction numerator 1 over denominator sin invisible function application theta end fraction

      • 1 plus tan squared invisible function application theta blank equals sec squared invisible function application theta blank

      • 1 plus cot squared invisible function application theta blank equals cosec squared invisible function application theta blank

      • sin invisible function application open parentheses A blank plus-or-minus B close parentheses equals sin invisible function application A cos invisible function application B blank plus-or-minus cos space A space sin invisible function application B

      • cos invisible function application open parentheses A blank plus-or-minus B close parentheses equals cos invisible function application A cos invisible function application B blank minus-or-plus sin space A space sin invisible function application B

      • tan invisible function application open parentheses A blank plus-or-minus B close parentheses equals fraction numerator tan invisible function application A blank plus-or-minus space tan invisible function application B over denominator 1 minus-or-plus tan invisible function application A tan invisible function application B end fraction

      • tan invisible function application 2 theta equals fraction numerator 2 tan invisible function application theta over denominator 1 minus tan squared invisible function application theta end fraction

    • The identity for cot is not in the formula booklet, you will need to remember it

      •  cot invisible function application theta equals fraction numerator 1 over denominator tan invisible function application theta end fraction equals fraction numerator cos invisible function application theta over denominator sin invisible function application theta end fraction

  • 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

What should I look out for when proving new trigonometric identities?

  • Look for anything that could be a part of one of the above identities on either side

    • For example if you see sin invisible function application 2 theta you can replace it with 2 sin invisible function application theta cos invisible function application theta blank

    • If you see 2 sin invisible function application theta cos invisible function application theta blankyou can replace it with sin invisible function application 2 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

  • Clever substitution into the compound angle formulae can be a useful tool for proving identities

    • For example rewriting cos space theta over 2 as cos space left parenthesis theta space minus space theta over 2 right parenthesis doesn’t change the ratio but could make an identity easier to prove

  • You will most likely need to be able to work with fractions and fractions-within-fractions

  • Always keep an eye on the 'target' expression – this can help suggest what identities to use

Examiner Tips and Tricks

  • Don't forget that you can start a proof from either end – sometimes it might be easier to start from the left-hand side and sometimes it may be easier to start from the right-hand side

  • Make sure you use the formula booklet as all of the relevant trigonometric identities are given to you 

  • Look out for special angles (0°, 90°, etc) as you may be able to quickly simplify or cancel parts of an expression (e.g. cos space 90 degree equals 0)

Worked Example

Prove that 8 cos to the power of 4 invisible function application theta minus 8 cos squared invisible function application theta plus 1 equals cos invisible function application 4 theta.

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Amber

Author: Amber

Expertise: Maths Content Creator

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.

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