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Graphs of trigonometric functions

This section describes the graphs of trigonometric functions.

Graph of sin θ

Graph of sin θ

Complete the following table, using your calculator to calculate the values. Then plot the values with sin θ on the y -axis and θ on the x -axis. Round answers to 1 decimal place.

θ 0 30 60 90 120 150
sin θ
θ 180 210 240 270 300 330 360
sin θ

Let us look back at our values for sin θ

θ 0 30 45 60 90 180
sin θ 0 1 2 1 2 3 2 1 0

As you can see, the function sin θ has a value of 0 at θ = 0 . Its value then smoothly increases until θ = 90 when its value is 1. We also know that it later decreases to 0 when θ = 180 . Putting all this together we can start to picture the full extent of the sine graph. The sine graph is shown in [link] . Notice the wave shape, with each wave having a length of 360 . We say the graph has a period of 360 . The height of the wave above (or below) the x -axis is called the wave's amplitude . Thus the maximum amplitude of the sine-wave is 1, and its minimum amplitude is -1.

The graph of sin θ .

Functions of the form y = a sin ( x ) + q

In the equation, y = a sin ( x ) + q , a and q are constants and have different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function f ( θ ) = 2 sin θ + 3 .

Graph of f ( θ ) = 2 sin θ + 3

Functions of the form y = a sin ( θ ) + q :

  1. On the same set of axes, plot the following graphs:
    1. a ( θ ) = sin θ - 2
    2. b ( θ ) = sin θ - 1
    3. c ( θ ) = sin θ
    4. d ( θ ) = sin θ + 1
    5. e ( θ ) = sin θ + 2
    Use your results to deduce the effect of q .
  2. On the same set of axes, plot the following graphs:
    1. f ( θ ) = - 2 · sin θ
    2. g ( θ ) = - 1 · sin θ
    3. h ( θ ) = 0 · sin θ
    4. j ( θ ) = 1 · sin θ
    5. k ( θ ) = 2 · sin θ
    Use your results to deduce the effect of a .

You should have found that the value of a affects the height of the peaks of the graph. As the magnitude of a increases, the peaks get higher. As it decreases, the peaks get lower.

q is called the vertical shift . If q = 2 , then the whole sine graph shifts up 2 units. If q = - 1 , the whole sine graph shifts down 1 unit.

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = a sin ( x ) + q .
a > 0 a < 0
q > 0
q < 0

Domain and range

For f ( θ ) = a sin ( θ ) + q , the domain is { θ : θ R } because there is no value of θ R for which f ( θ ) is undefined.

The range of f ( θ ) = a sin θ + q depends on whether the value for a is positive or negative. We will consider these two cases separately.

If a > 0 we have:

- 1 sin θ 1 - a a sin θ a ( Multiplication by a positive number maintains the nature of the inequality ) - a + q a sin θ + q a + q - a + q f ( θ ) a + q

This tells us that for all values of θ , f ( θ ) is always between - a + q and a + q . Therefore if a > 0 , the range of f ( θ ) = a sin θ + q is { f ( θ ) : f ( θ ) [ - a + q , a + q ] } .

Similarly, it can be shown that if a < 0 , the range of f ( θ ) = a sin θ + q is { f ( θ ) : f ( θ ) [ a + q , - a + q ] } . This is left as an exercise.

The easiest way to find the range is simply to look for the "bottom" and the "top" of the graph.


The y -intercept, y i n t , of f ( θ ) = a sin ( x ) + q is simply the value of f ( θ ) at θ = 0 .

y i n t = f ( 0 ) = a sin ( 0 ) + q = a ( 0 ) + q = q

Graph of cos θ

Graph of cos θ :

Complete the following table, using your calculator to calculate the values correct to 1 decimal place. Then plot the values with cos θ on the y -axis and θ on the x -axis.

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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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