Solving Trigonometric Equations (DP IB Maths: AI HL)

What are the graphs of trigonometric functions?

• The trigonometric functions sin, cos and tan all have special periodic graphs
• You’ll need to know their properties and how to sketch them for a given domain in either degrees or radians
• Sketching the trigonometric graphs can help to
  • Solve trigonometric equations and find all solutions
  • Understand transformations of trigonometric functions

 

What are the properties of the graphs of sin x and cos x?

• The graphs of sin x and cos x are both periodic
  • They repeat every 360° (2π radians)
  • The angle will always be on the x-axis
    • Either in degrees or radians
• The graphs of sin x and cos x are always in the range -1 ≤ y ≤ 1
  • Domain: 
  • Range: 
  • The graphs of sin x and cos x are identical however one is a translation of the other
    • sin x passes through the origin
    • cos x passes through (0, 1)
• The amplitude of the graphs of sin x and cos x is 1

What are the properties of the graph of tan x?

• The graph of tan x is periodic
  • It repeats every 180° (π radians)
  • The angle will always be on the x-axis
    • Either in degrees or radians
• The graph of tan x is undefined at the points ± 90°, ± 270° etc
  • There are asymptotes at these points on the graph
  • In radians this is at the points ± , ± etc
• The range of the graph of tan x is
  • Domain:  
  • Range: 

How do I sketch trigonometric graphs?

• You may need to sketch a trigonometric graph so you will need to remember the key features of each one
• The following steps may help you sketch a trigonometric graph
  • STEP 1: Check whether you should be working in degrees or radians
    • You should check the domain given for this
    • If you see π in the given domain then you should work in radians
  • STEP 2: Label the x-axis in multiples of 90°
    • This will be multiples of if you are working in radians
    • Make sure you cover the whole domain on the x-axis
  • STEP 3: Label the y-axis
    • The range for the y-axis will be – 1 ≤ y ≤ 1 for sin or cos
    • For tan you will not need any specific points on the y-axis
  • STEP 4: Draw the graph
    • Knowing exact values will help with this, such as remembering that sin(0) = 0 and
      cos(0) = 1
    • Mark the important points on the axis first
    • If you are drawing the graph of tan x put the asymptotes in first
    • If you are drawing sin x or cos x mark in where the maximum and minimum points will be
    • Try to keep the symmetry and rotational symmetry as you sketch, as this will help when using the graph to find solutions

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)