IBMaths: AA HLDPRevision Notes3. Geometry & Trigonometry3.8 Further Trigonometry3.8.2 Strategy for Trigonometric Equations

Strategy for Trigonometric Equations (DP IB Maths: AA HL)

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Strategy for Trigonometric Equations

How do I approach solving trig equations?

  • You can solve trig equations in a variety of different ways
    • Sketching a graph 
      • If you have your GDC it is always worth sketching the graph and using this to analyse its features
    • Using trigonometric identities, Pythagorean identities, the compound or double angle identities
      • Almost all of these are in the formula booklet, make sure you have it open at the right page
    • Using the unit circle
    • Factorising quadratic trig equations 
      • Look out for quadratics such as 5tan2x – 3tan x – 4 = 0
  • The final rearranged equation you solve will involve sin, cos or tan
    • Don’t try to solve an equation with cosec, sec, or cot directly

What should I look for when solving trig equations?

  • Check the value of x or θ
    • If it is just x or θ you can begin solving
    • If there are different multiples of x or θ you will need to use the double angle formulae to get everything in terms of the same multiple of x or θ
    • If it is a function of x or θ, e.g. 2x – 15, you will need to transform the range first
      • You must remember to transform your solutions back again at the end
  • Does it involve more than one trigonometric function?
    • If it does, try to rearrange everything to bring it to one side, you may need to factorise
    • If not, can you use an identity to reduce the number of different trigonometric functions?
      • You should be able to use identities to reduce everything to just one simple trig function (either sin, cos or tan)
  • Is it linear or quadratic?
    • If it is linear you should be able to rearrange and solve it
    • If it is quadratic you may need to factorise first
      • You will most likely get two solutions, consider whether they both exist
      • Remember solutions to sin x = k and cos x = k only exist for -1 ≤ k ≤ 1 whereas solutions to tan x = k exist for all values of k
  • Are my solutions within the given range and do I need to find more solutions?
    • Be extra careful if your solutions are negative but the given range is positive only
    • Use a sketch of the graph or the unit circle to find the other solutions within the range
    • If you have a function of x or θ make sure you are finding the solutions within the transformed range
      • Don’t forget to transform the solutions back so that they are in the required range at the end

Strategy for Further Trigonometric Equations Diagram 1

Examiner Tip

  • Try to use identities and formulas to reduce the equation into its simplest terms.
  • Don’t forget to check the function range and ensure you have included all possible solutions.
  • If the question involves a function of x or θ ensure you transform the range first (and ensure you transform your solutions back again at the end!).

Worked example

Find the solutions of the equation open parentheses 1 plus cot squared invisible function application 2 theta close parentheses open parentheses 5 cos squared invisible function application theta minus 1 close parentheses equals cot squared invisible function application 2 theta blankin the interval 0 blank less or equal than blank theta blank less or equal than 2 pi.

3-8-2-ib-aa-hl-equation-strategy-we-solution

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    Amber

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