Graphs of Trigonometric Functions (CIE IGCSE Additional Maths)

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 do I use trigonometric graphs?

• By sketching the graph you can read off all the solutions in a given range (or interval)
• Your calculator will only give you the principal 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

• You should recognise any values/angles associated with exact values
• You should be able to spot the pattern of solutions using the symmetry and periodicity of the graph

 

What transformations of trigonometric functions do I need to know?

• As with other graphs of functions, trigonometric graphs can be transformed through translations, stretches and reflections
• Translations can be either horizontal (parallel to the x-axis) or vertical (parallel to the y-axis)
  • For the function y = sin (x)
    • A vertical translation of a units in the positive direction (up) is denoted by
      y = sin (x) + a
    • A vertical translation of a units in the negative direction (down) is denoted by
      y = sin (x) - a
    • A horizontal translation in the positive direction (right) is denoted by y = sin (x - a)
    • A horizontal translation in the negative direction (left) is denoted by y = sin (x + a)
• Stretches can be either horizontal (parallel to the x-axis) or vertical (parallel to the y-axis)
  • For the function y = sin (x)
    • A vertical stretch of a factor a units is denoted by y = a sin (x)
    • A horizontal stretch of a factor a units is denoted by y = sin ()
• Reflections can be either across the x-axis or across the y-axis
  • For the function y = sin (x)
    • A reflection across the x-axis is denoted by y = - sin (x)

 

What combined transformations are there?

• Stretches in the horizontal and vertical direction are often combined
• The functions a sin(bx) and a cos(bx) have the following properties:
  • The amplitude of the graph is |a |
  • The period of the graph is ° (or rad)
    • In this course, a will always be a positive integer and b will be a simple fraction or integer
• Translations in both directions could also be combined with the stretches
• The functions a sinbx + c and a cosbx + c have the following properties:
  • The amplitude of the graph is |a |
  • The period of the graph is ° (or )
  • The translation in the vertical direction is c
    • c represents the principal axis (the line that the function fluctuates about)
    • It helps to start by sketching the principle axis
• The function a tanbx + c has the following properties:
  • The amplitude of the graph does not exist
  • The period of the graph is ° (or )
  • The translation in the vertical direction (principal axis) is c
    • Finding and drawing the asymptotes first can help to sketch these graphs 

How do I sketch transformations of trigonometric functions?

• Sketch the graph of the original function first
• Carry out each transformation separately
  • The order in which you carry out the transformations is important
  • Given the form y = a sinbx + c carry out any stretches first, translations next and reflections last
  • Use a very light pencil to mark where the graph has moved for each transformation
• It is a good idea to mark in the principal axis the lines corresponding to the maximum and minimum points first
  • The principal axis will be the line y = c
  • The maximum points will be on the line y = c + a
  • The minimum points will be on the line y = c - a
• Sketch in the new transformed graph
• Check it is correct by looking at some key points from the exact values