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Let us look back at our values for cos θ

θ 0 30 45 60 90 180
cos θ 1 3 2 1 2 1 2 0 - 1

If you look carefully, you will notice that the cosine of an angle θ is the same as the sine of the angle 90 - θ . Take for example,

cos 60 = 1 2 = sin 30 = sin ( 90 - 60 )

This tells us that in order to create the cosine graph, all we need to do is to shift the sine graph 90 to the left. The graph of cos θ is shown in [link] . As the cosine graph is simply a shifted sine graph, it will have the same period and amplitude as the sine graph.

The graph of cos θ .

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

In the equation, y = a cos ( 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 cos θ + 3 .

Graph of f ( θ ) = 2 cos θ + 3

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

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

You should have found that the value of a affects the amplitude of the cosine graph in the same way it did for the sine graph.

You should have also found that the value of q shifts the cosine graph in the same way as it did the sine graph.

These different properties are summarised in [link] .

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

Domain and range

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

It is easy to see that the range of f ( θ ) will be the same as the range of a sin ( θ ) + q . This is because the maximum and minimum values of a cos ( θ ) + q will be the same as the maximum and minimum values of a sin ( θ ) + q .


The y -intercept of f ( θ ) = a cos ( x ) + q is calculated in the same way as for sine.

y i n t = f ( 0 ) = a cos ( 0 ) + q = a ( 1 ) + q = a + q

Comparison of graphs of sin θ And cos θ

The graph of cos θ (solid-line) and the graph of sin θ (dashed-line).

Notice that the two graphs look very similar. Both oscillate up and down around the x -axis as you move along the axis. The distances between the peaks of the two graphs is the same and is constant along each graph. The height of the peaks and the depths of the troughs are the same.

The only difference is that the sin graph is shifted a little to the right of the cos graph by 90 . That means that if you shift the whole cos graph to the right by 90 it will overlap perfectly with the sin graph. You could also move the sin graph by 90 to the left and it would overlap perfectly with the cos graph. This means that:

sin θ = cos ( θ - 90 ) ( shift the cos graph to the right ) a nd cos θ = sin ( θ + 90 ) ( shift the sin graph to the left )

Graph of tan θ

Graph of tan θ

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

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

Let us look back at our values for tan θ

Questions & Answers

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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Maths grade 10 rought draft. OpenStax CNX. Sep 29, 2011 Download for free at http://cnx.org/content/col11363/1.1
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