Periodic functions > The sine function
123456The sine function

Theory

Take a look at the applet: sine function

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

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

Because of the period of 2 π all solutions of sin ( x ) = c are given by :
x = arcsin ( c ) + k 2 π x = π arcsin ( c ) + k 2 π with k any integer.

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

There are some values that are convenient to use:

  • sin ( 0 ) = 0

  • sin ( 1 6 π ) = 1 2

  • sin ( 1 4 π ) = 1 2 2

  • sin ( 1 3 π ) = 1 2 3

  • sin ( 1 2 π ) = 1

and vice versa:

  • arcsin ( 0 ) = 0

  • arcsin ( 1 2 ) = 1 6 π

  • arcsin ( 1 2 2 ) = 1 4 π

  • arcsin ( 1 2 3 ) = 1 3 π

  • arcsin ( 1 ) = 1 2 π

You should use these values if exact values are required.

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