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

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 ∈  

  • 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 ∈  

  • 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 ∈   

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