Strategy for Trigonometric Equations (DP IB Analysis & Approaches (AA)): Revision Note

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