Sine

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

$\sin(\theta )={\frac {\mbox{opposite}}{\mbox{hypotenuse}}}$

$\text{Opposite}$ 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:

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

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 $-\cos(x)$ .

Addition of angles under sine:

$\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$

$\sin(2a)=2\sin(a)\cos(a)$

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

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

The square of sine:

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

See also

• Cosine
• Cosecant
• Hyperbolic sine
• Law of sines