Sine () is a trigonometric ratio. In a right triangle with an angle ,

is the side of the triangle facing(opposite to) angle , and is the side opposite the right angle.

## Properties

The sine of an angle is the y-coordinate of the point of intersection of said angle and a unit circle.

As a result of Euler's formula, the sine function can also be represented as

[Proof]

If desired, the sine function may be calculated as a direct summation series:

The reciprocal of sine is cosecant (abbreviated as ), while its inverse is or . Note that sine is not being raised to the power of -1; this is an inverse function, not a reciprocal.

The derivative of is , while its antiderivative is . The derivative of is

### Trigonometric identities

Sine and cosine can be converted between each other.

Proof: Angle_Sum_for_Sine

[Proof]

The sine of an imaginary number becomes a variant of a hyperbolic sine:

The square of sine, and half angle theorem:

Proofs

### Approximations

For small values of , there is an easy approximation: