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# 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 : sinB : 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 : sinB : 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.