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Sine

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

\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}

\text{Opposite} is the side of the triangle facing(opposite to) angle \theta , and \text{hypotenuse} 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

\sin(\theta)=\frac{e^{\theta i}-e^{-\theta i}}{2i}

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

\sin(\theta)=\sum_{k=0}^\infty\frac{(-1)^k x^{2k+1}}{(2k+1)!}

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

The derivative of \sin(x) is \cos(x) , while its antiderivative is -\cos(x) .

Trigonometric identities

Sine and cosine can be converted between each other.

\sin(x)=\cos(\frac{\pi}{2}-x)=\cos(x-\frac{\pi}{2})=- \cos(\frac{\pi}{2}+x)
\sin^2(x)+\cos^2(x) = 1

Addition of angles under sine:

\begin{align}&\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)\\
&\sin(2a)=2\sin(a)\cos(a)\end{align}

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

\sin(\theta i)=i\sinh(\theta)

The square of sine:

\sin^2(\theta)=\frac{1-\cos(2\theta)}{2}

Approximations

For small values of \theta, you can use one iteration in the taylor series:

\sin \theta \approx \theta \mbox{ if } \theta < 0.5

See also

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