# sine

The graph of *y* = sin *x*.

The sine function, sin *a*, is a trigonometric
functions, the ratio of the opposite side to the hypotenuse in a right-angled triangle.

For the graph of *y* = sin *x*, shown in the diagram, if the *x*-axis is extended, the graph repeats itself every 360°, i.e. the
period is 360°. The amplitude oscillates between +1 and -1.

## The sine rule or theorem

In **plane geometry**, if *a*, *b*, and *c* are the sides, and *A*, *B*, and *C* the angles of
a triangle, then *a* : *b *: *c* = sin *A* : sin*B* : sin *C*. In other words:

*a*/sin *A* = *b*/sin *B* = *c*/sin *C*.

The theorem can be used to solve a triangle if one side, the opposite angle, and a further side or a further angle are given.

In **spherical geometry**, if *a*, *b*, and *c* are the sides, and *A*, *B*, and *C* the angles of
a spherical triangle, then sin *a* : sin *b* : sin *c* = sin *A* : sin*B* : sin *C*. In other words:

sin *a*/sin *A* = sin *b*/sin *B* = sin *c*/sin *C*.

The sine theorem of spherical geometry, like that of plane geometry, can be used to solve a spherical triangle if one side, the opposite angle, and a further side or a further angle are given.