Solving Equations Using Trigonometric Graphs (DP IB Analysis & Approaches (AA)): Revision Note

Using Trigonometric Graphs

How can I use a trigonometric graph to find extra solutions?

• Your calculator will only give you the first solution to a problem such as sin^-1(0.5)

  • This solution is called the primary value

• However, due to the periodic nature of the trig functions there could be an infinite number of solutions

  • Further solutions are called the secondary values

• This is why you will be given a domain (interval) in which your solutions should be found

  • This could either be in degrees or in radians

    • If you see π or some multiple of π then you must work in radians

• The following steps will help you use the trigonometric graphs to find secondary values

  • STEP 1: Sketch the graph for the given function and interval

    • Check whether you should be working in degrees or radians and label the axes with the key values

  • STEP 2: Draw a horizontal line going through the y-axis at the point you are trying to find the values for

    • For example if you are looking for the solutions to sin^-1(-0.5) then draw the horizontal line going through the y-axis at -0.5

    • The number of times this line cuts the graph is the number of solutions within the given interval

  • STEP 3: Find the primary value and mark it on the graph

    • This will either be an exact value and you should know it

    • Or you will be able to use your calculator to find it

  • STEP 4: Use the symmetry of the graph to find all the solutions in the interval by adding or subtracting from the key values on the graph

What patterns can be seen from the graphs of trigonometric functions?

• The graph of sin x has rotational symmetry about the origin

  • So sin(-x) = - sin(x)

  • sin(x) = sin(180° - x) or sin(π – x)

• The graph of cos x has reflectional symmetry about the y-axis

  • So cos(-x) = cos(x)

  • cos(x) = cos(360° – x) or cos(2π – x)

• The graph of tan x repeats every 180° (π radians)

  • So tan(x) = tan(x ± 180°) or tan(x ±  π )

• The graphs of sin x and cos x repeat every 360° (2π radians)

  • So sin(x) = sin(x ±  360°) or sin(x  ±  2π)

  • cos(x) = cos(x ±  360°) or cos(x  ±  2π)