IBMaths: AA HLDPRevision Notes3. Geometry & Trigonometry3.6 Trigonometric Equations & Identities3.6.5 Linear Trigonometric Equations

Linear Trigonometric Equations (DP IB Maths: AA HL)

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Trigonometric Equations: sinx = k

How are trigonometric equations solved?

  • Trigonometric equations can have an infinite number of solutions
    • For an equation in sin or cos you can add 360° or 2π to each solution to find more solutions
    • For an equation in tan you can add 180° or π to each solution
  • When solving a trigonometric equation you will be given a range of values within which you should find all the values
  • Solving the equation normally and using the inverse function on your calculator or your knowledge of exact values will give you the primary value
  • The secondary values can be found with the help of:
    • The unit circle
    • The graphs of trigonometric functions

How are trigonometric equations of the form sin x = k solved?

  • It is a good idea to sketch the graph of the trigonometric function first
    • Use the given range of values as the domain for your graph
    • The intersections of the graph of the function and the line y = k will show you
      • The location of the solutions
      • The number of solutions
    • You will be able to use the symmetry properties of the graph to find all secondary values within the given range of values
  • The method for finding secondary values are:
    • For the equation sin x = k the primary value is x1 = sin -1 k
      • A secondary value is x2 = 180° - sin -1 k
      • Then all values within the range can be found using x1 ± 360n and
        x2 ± 360n where n ∈  straight natural numbers
    • For the equation cos x = k the primary value is x1 = cos -1 k 
      • A secondary value is x2 = - cos -1 k
      • Then all values within the range can be found using x1 ± 360n and
        x2 ± 360n where n ∈  straight natural numbers
    • For the equation tan x = k  the primary value is x = tan -1 k
      • All secondary values within the range can be found using x ± 180n where n ∈  straight natural numbers 

Examiner Tip

  • If you are using your GDC it will only give you the principal value and you need to find all other solutions for the given interval
  • Sketch out the CAST diagram and the trig graphs on your exam paper to refer back to as many times as you need to

Worked example

Solve the equation 2 cos space x space equals space minus 1 , finding all solutions in the range negative pi space less or equal than space x space less or equal than space pi.

aa-sl-3-6-4-trig-equations-sinx--k-we-solution

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Trigonometric Equations: sin(ax + b) = k

How can I solve equations with transformations of trig functions?

  • Trigonometric equations in the form sin(ax + b) can be solved in more than one way
  • The easiest method is to consider the transformation of the angle as a substitution
    • For example let u = ax + b
  • Transform the given interval for the solutions in the same way as the angle
    • For example if the given interval is 0° ≤ x ≤ 360° the new interval will be
    • (a (0°) + b) ≤ u ≤ (a (360°) + b)
  • Solve the function to find the primary value for u
  • Use either the unit circle or sketch the graph to find all the other solutions in the range for u
  • Undo the substitution to convert all of the solutions back into the corresponding solutions for x
  • Another method would be to sketch the transformation of the function
    • If you use this method then you will not need to use a substitution for the range of values

Examiner Tip

  • If you transform the interval, remember to convert the found angles back to the final values at the end!
  • If you are using your GDC it will only give you the principal value and you need to find all other solutions for the given interval
  • Sketch out the CAST diagram and the trig graphs on your exam paper to refer back to as many times as you need to

Worked example

Solve the equation 2 cos left parenthesis 2 x space minus space 30 degree right parenthesis space equals space minus 1, finding all solutions in the range negative 360 degree space less or equal than space x space less or equal than space 360 degree.

aa-sl-3-6-4-trig-equations-sinaxb--k-we-solution

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    Amber

    Author: Amber

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    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.