Trigonometric Graphs (AQA GCSE Further Maths): Revision Note

Graphs of trigonometric functions

What is meant by graphs of trigonometric functions?

• The graphs are

  • $y=\sin x$

  • $y=\cos x$

  • $y=\tan x$

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°

Properties of trigonometric graphs

What is meant by the properties of trigonometric graphs?

• Properties refers to any special features or patterns

  • The graphs $y=\sin x,y=\cos x$ and $y=\tan x$ are all periodic

    • This means their graphs/curves repeat every so often

    • The length of the interval after which the graph repeats 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 $y=\sin x$ ($-1\le y\le 1$)

• The graph $y=\sin x$ repeats every 360° (has period 360)

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

• 'Something' happens every 90°

  • $y=\sin x$ cycles between 0, 1, 0, -1, 0, 1, 0, -1, ...

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

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

  • $y=\sin x$ has rotational symmetry around the origin

  • The graph has an amplitude of 1

    • Amplitude means the height of the sine graph above zero

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

• The graph of $y=\cos x$ is almost the same as the graph of $y=\sin x$

  • 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 $y=\cos x$ ($-1\le y\le 1$)

• The graph $y=\cos x$ repeats every 360° (has period 360)

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

• 'Something' happens every 90°

  • $y=\cos x$ cycles between 1, 0, -1, 0, 1, 0, -1, ...

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

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

  • $y=\cos x$ has a line of symmetry in the y-axis (the line with equation $x=0$)

  • The graph has an amplitude of 1

    • Amplitude means the height of the cosine graph above zero

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 $y=\tan x$ 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°

  • $y=\tan x$ cycles between 0, asymptote, 0, asymptote, 0, ...

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

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

  • $y=\tan x$ has rotational symmetry around the origin