Trigonometric Graphs (AQA GCSE Further Maths) : Revision Note

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Reviewed by: Dan Finlay

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Graphs of Trigonometric Functions

What is meant by graphs of trigonometric functions?

  • The graphs are

    • y equals sin space x

    • y equals cos space x

    • y equals tan space x

Why do I need to know what graphs of trigonometric functions look like?

  • Trigonometric graphs (trig graphs) are used in various applications of mathematics

    • e.g.  the oscillating / wave-like nature of sine and/or cosine can be used to model how a pendulum swings or tide heights

How do I sketch trig graphs?

  • As with other graphs, being familiar with the general style of trig graphs will help you sketch them quickly

    • They can then be used to find values or angles alongside, or instead of, your calculator

  • All trig graphs follow a pattern – a “starting point” and then “something happens every 90°”

  • The diagrams below show the graphs of sin, cos and tan from -360° to 360°

    • Most questions will focus on the postive part of the graph for angles between 0° and 360°

y-equals-sinx
y-equals-cosx
y-equals-tanx

Worked Example

On the axes provided, sketch the graph of y equals sin space x degree for 0 less or equal than x less or equal than 360.

Mark key values on the axes provided; 1and −1 on the y-axis and 90, 180, 270 and 360 on the x-axis
Try to space them evenly apart but also remember this is a sketch! 

IcJiYEqM_2-14-trig-graphs1

 y equals sin space x degree starts at (0, 0) then every 90° it cycles though 1, 0 , −1, 0, ...
Mark these points on the axes

Finally, join the points with a smooth curve
You will get better at this with practice but again remember it is a sketch so do not spend ages making it look perfect!
It is best practice to label the curve with its equation

5_-KJTZ8_2-14-trig-graphs2

Properties of Trigonometric Graphs

What is meant by the properties of trigonometric graphs?

  • Properties refers to any special features or patterns

    • The graphs y equals sin space x comma space y equals cos space x and y equals tan space x are all periodic

      • This means their graphs/curves repeat every so often

      • The frequency (rate) at which they repeat is called the period

What are the properties of the graph y = sin x?

  • Angles will always be on the x-axis

  • Values of sine run between -1 and 1

    • So the y-axis will only need to run between -1 and 1

      • You may see this referred to as the range of y equals sin space x (negative 1 less or equal than y less or equal than 1)

  • The graph y equals sin space x repeats every 360° (has period 360)

  • The graph 'starts' at (0, 0) (but can have negative angles too)

  • 'Something' happens every 90°

    • y equals sin space x cycles between 0, 1, 0, -1, 0, 1, 0, -1, ...

      • ... every 90° starting at (0, 0)

  • Other properties of the the graph of y equals sin space x that can be helpful but do not crop up often include

    • y equals sin space x has rotational symmetry around the origin

    • The graph has an amplitude of 1

      • Amplitude means the height of the sine graph above zero

5-2-1-graphs-of-trigonometric-functions-diagram-v3tan5-2-1-graphs-of-trigonometric-functions-diagram-v1sinx-1

 

What are the properties of the graph y = cos x?

  • The graph of y equals cos space x is almost the same as the graph of y equals sin space x

    • but it has a different starting point

    • at (0, 1)

  • Angles will always be on the x-axis

  • Values of cosine run between -1 and 1

    • So the y-axis will only need to run between -1 and 1

      • You may see this referred to as the range of y equals cos space x (negative 1 less or equal than y less or equal than 1)

  • The graph y equals cos space x repeats every 360° (has period 360)

  • The graph 'starts' at (0, 1) (but can have negative angles too)

  • 'Something' happens every 90°

    • y equals cos space x cycles between 1, 0, -1, 0, 1, 0, -1, ...

      • ... every 90° starting at (0, 1)

  • Other properties of the the graph of y equals cos space x that can be helpful but do not crop up often include

    • y equals cos space x has a line of symmetry in the y-axis (the line with equation x equals 0)

    • The graph has an amplitude of 1

      • Amplitude means the height of the cosine graph above zero

5-2-1-graphs-of-trigonometric-functions-diagram-v2-cosx

What are the properties of the graph of y = tan x?

  • Angles will always be on the x-axis

  • Values of tangent have no limit

    • So in general the y-axis will not need to have any labels on it

      • but a label referring to a particular value in a question may be helpful

  • The graph y equals tan space x repeats every 180° (has period 180)

  • The graph 'starts' at (0, 0) (but can have negative angles too)

  • The graph has asymptotes (also known as discontinuities) which are vertical lines that the graph approaches

    • but never intersects (touches or crosses)

    • asymptotes are drawn as dotted vertical lines on graphs

    • the asymptote arises when trying to divide by zero (which is undefined)

  • 'Something' happens every 90°

    • y equals tan space x cycles between 0, asymptote, 0, asymptote, 0, ...

      • ... every 90° starting at (0, 0)

  • Other properties of the the graph of y equals tan space x that can be helpful but do not crop up often include

    • y equals tan space x has rotational symmetry around the origin

5-2-1-graphs-of-trigonometric-functions-diagram-v3tan

 

Examiner Tips and Tricks

  • Sketch all three trig graphs on your exam paper so you can refer to them as many times as you need to!

Worked Example

The graph of y equals sin space x between 0° and 360° passes through the point with coordinates open parentheses 30 comma space 0.5 close parentheses.

Using a symmetry argument, find the value of k such that open parentheses k comma space 0.5 close parentheses is a different point on graph.

 

Sketch the graph of y = sin x and mark on the point where x = 30 and y = 0.5

cie-igcse-3-12-2-solving-trig-equations-1

On the graph, mark on another point with the same y-height of 0.5
By symmetry of the curve between 0° and 180°, this will have an x-coordinate of 180 - 30

the coordinates are (180 - 30, 0.5) 

The value of k is 180 - 30 = 150

k = 150

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Paul

Author: Paul

Expertise: Maths Content Creator (Previous)

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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