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

Using trigonometric graphs

How do I solve trig equations?

• The inverse trig functions on your GDC will only give you one solution to a trig equation

  • This solution is called the principal value

    • e.g. ${\sin }^{-1}\left(0.7\right)$ is the principal value for the equation $\sin x=0.7$

• You can use the trig graphs and their symmetries to find the other solutions within an interval

How do I determine the number of solutions?

• For $\sin x=k$ and $\cos x=k$ where $-1<k<1$ and $k\ne 0$

  • There are always two solutions in any interval of length 360°

  • Divide the width of the interval by 360°

    • If it is a whole number then double it to get the number of solutions

    • Otherwise, double the closest whole numbers to find the minimum and maximum number of solutions

• For $\tan x=k$

  • There is always one solution in any interval of length 180°

  • Divide the width of the interval by 180°

    • If it is a whole number then this is equal to the number of solutions

    • Otherwise, the closest whole numbers are the minimum and maximum number of solutions

How do I use trig graphs to solve trig equations?

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

  • Check whether you should be working in degrees or radians

  • Label the axes with the key values (0°, 90°, 180°, etc)

• STEP 2
  Draw a horizontal line going through the y-axis at the relevant point

  • e.g. to solve $\sin x=0.7$ draw the line $y=0.7$

• STEP 3
  Find the principal value and mark it on the graph

• STEP 4
  Use the symmetry and periodicity of the graph to find all the solutions in the interval

  • $y=\sin x$ is symmetrical about $x=90°$ and repeats every 360°

    • If $x=50°$ a solution, then $x=180°-50°$ is also a solution

  • $y=\cos x$ is symmetrical about $x=0°$ and repeats every 360°

    • If $x=50°$ a solution, then $x=-50°$ is also a solution

  • $y=\tan x$ repeats every 180°

    • If $x=50°$ a solution, then $x=50°+180°$ is also a solution