Graphs of Trigonometric Functions (Cambridge (CIE) O Level Additional Maths): Revision Note

Graphs of trig functions

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: $\left\{\mathit{x}\mathit{ }|\mathit{ }\mathit{x}\mathbb{ }\in \mathbb{ }\mathrm{ℝ}\right\}$

  • Range: $\left\{\mathit{y}\mathit{ }|-1\mathit{ }\le \mathit{y}\mathit{ }\le 1\right\}$

  • 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 ± $\frac{\mathrm{\pi }}{2}$, ± $\frac{3\mathrm{\pi }}{2}$ etc

• The range of the graph of tan x is

  • Domain: $\left\{\mathit{x}\mathit{ }|\mathit{ }\mathit{x}\mathit{ }\ne \frac{\mathit{\pi }}{2}+\mathit{k}\mathit{\pi },\mathit{ }\mathit{ }\mathit{k}\mathbb{ }\in \mathbb{ }\mathrm{\mathbb{Z}}\right\}$ 

  • Range: $\left\{\mathit{y}\mathit{ }|\mathit{ }\mathit{y}\mathbb{ }\in \mathbb{ }\mathrm{ℝ}\right\}$

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 $\frac{\mathrm{\pi }}{2}$ 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

Transformations of trig functions

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 ($\frac{x}{a}$)

• 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 $\frac{360}{b}$° (or $\frac{2\mathrm{\pi }}{b}$ 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 $\frac{360}{b}$° (or $\frac{2\pi }{b}$)

  • 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 $\frac{180}{b}$° (or $\frac{\mathit{\pi }}{\mathit{b}}$)

  • 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