Graphs of Trigonometric Functions (DP IB Applications & Interpretation (AI)): Revision Note

Graphs of Trigonometric 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: 

  • 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