Periodic functions > The cosine function
123456The cosine function

Theory

Take a look at the applet: Cosine function

Above you see the graph of f ( x ) = cos ( x ) with x in radians on [ 0 , 2 π ] . The solutions of cos ( x ) = c are shown ( c is a constant).

The solution of cos ( x ) = c within [ 1 2 π , 1 2 π ] is arccosine of c: x = arccos ( c ) .
There (often) is another solution within a range of one period.
Due to the symmetry of the graph that other solution is x = - arccos ( c ) .

Because of the period of 2 π all solutions of cos ( x ) = c are given by:
x = arccos ( x ) + k 2 π x = - arccos ( x ) + k 2 π where k k is any integer.

The equation cos ( x ) = c only has solutions if -1 c 1 .

The graph of the cosine function strongly resembles the graph of the sine function. Therefore there are several relations between the two.

Take a look at the applet: Unit circle

The graph of f ( x ) = cos ( x ) with x in radians, the standard cosine graph strongly resembles the standard sine graph and the period also is 2 π . It has only been shifted to the left by 1/2Π.

This means that cos ( x ) = sin ( x + 1 2 π ) .

Furthermore, it is possible to draw the sine and cosine of the same angle in one unit circle. When you do this, you see that the coördinates op point P can be written as: x P = cos ( α ) and y P = sin ( α ) .
Using Pythagoras' theorem you can check that:
( sin ( α ) ) 2 + ( cos ( α ) ) 2 = 1
To decrease the number of brackets you can use:
sin 2 ( α ) + cos 2 ( α ) = 1

There are some values that are easy to use:

  • cos ( 0 ) = 1

  • cos ( 1 6 π ) = 1 2 3

  • cos ( 1 4 π ) = 1 2 2

  • cos ( 1 3 π ) = 1 2

  • cos ( 1 2 π ) = 0

and vice versa:

  • arccos ( 0 ) = 1 2 π

  • arccos ( 1 2 ) = 1 3 π

  • arccos ( 1 2 2 ) = 1 4 π

  • arccos ( 1 2 3 ) = 1 6 π

  • arccos ( 1 ) = 1

Use these values, when exact answers are required.

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