Sine

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

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:

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

Limits

$\lim_{x\to 0} \left(\frac{\sin (x)}{x}\right) = 1$

Approximations

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

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

See also

• Cosine
• Cosecant
• Hyperbolic sine
• Law of sines