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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.

PropertiesEdit

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) $ . The derivative of $ \arcsin(x) $ is $ \frac{1}{\sqrt{1-x^2}} $

Trigonometric identitiesEdit

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} $

LimitsEdit

$ \lim_{x\to 0} \frac{sin x}{x} = 1 $

ApproximationsEdit

For small values of $ \theta $, there is an easy approximation:

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

See alsoEdit