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
Addition of angles under sine:
- [Proof]
The sine of an imaginary number becomes a variant of a hyperbolic sine:
The square of sine, and half angle theorem:
Limits[]
Approximations[]
For small values of , there is an easy approximation:
See also[]
- Cosine
- Cosecant
- Hyperbolic sine
- Law of sines