In trigonometry, the sine of an angle equals the y-coordinate of the point where a line drawn at that angle through the center of a coordinate grid intersects a unit circle.

            _____|_____     _
           /     |   /|\     \
          /      |  / | \     } sine
         |       | /  |  |  _/
         |       |       |
         |       |       |
          \      |      /

In general, it is defined (in a right triangle) as the ratio of the length of the side opposite the angle to the length of the hypotenuse; in the unit circle diagram, however, the hypotenuse is always one unit in length.

                   / |
                  /  |
     hypotenuse  /   |  opposite
                /    |
               /     |
              /     _|

The name is derived from the Latin word sinus, meaning "curve". The sine of a number is just another number, of course, but when the sine function is graphed on a coordinate graph, it produces a smooth, rolling ("sinusoidal") curve, hence the name.

Sine is an elliptic curve. More to the point, any particular sin curve is the exact replica of the edge of a particular ellipse. Observe the sin curve for y=sin(x), 0<x<2*pi:

| 1                  ,,oO```Oo,,
|                 ,oO`         `Oo,
|               ,/`               `\,
|             ,/`                   `\,
|           ,/`                       `\,
|          /`                           `\
|        ,/                               \,
|       /`                                 `\
|     ,/                                     \,   pi
|    /`                                       `\   |
|  ,/                                           \, |
|,/`                                             `\,                    3*pi/2
|                        pi/2                        \,                                       ,/
|                                                     `\                                     /`
|                                                       \,                                 ,/
|                                                        `\                               /`
|                                                          \,                           ,/
|                                                           `\,                       ,/`
|                                                             `\,                   ,/`
|                                                               `\,               ,/`
|                                                                 `Oo,         ,oO`
| -1                                                                 ``Oo,,,oO``

Imagine that the straight horizontal line (above) is the path a circle of radius one follows as it rolls to the right. Then imagine that, instead of just a circle, it's a cylinder of radius one and length two. (The cylinder is rolling 'on top of' the entire picture above, as if you had it in your hand and were rolling it on your screen.) Now consider an ellipse on the surface of the cylinder: going from one 'corner' to the other. As the cylinder rolls, the ellipse will trace the sine curve above, and a circle at the midpoint of the cylinder will trace the straight line. Thus, whenever you are calculating anything with sin or cos, remember that you are actually using ellipses.

One series for finding sin(x) is

sin(x) |-> x/1! - x3/3! + x5/5! - x7/7! +...

It is obvious from this that sin(0) = 0, since all the terms evaluate to 0.

If infinite series are not your thing, try this infinite product, although it is not very clear in HTML:

sin(x) = (1 - x22 ) * (1 - x2(22π2)) * (1 - x2/(32π2))* ...

Note that these definitions of sine require an input in radians, not degrees, with which some are more familiar. To convert to degrees, simply multiply by 180/pi.

Like cosine, sine is periodic in 2π. The derivative of sine is cosine, and its integral is -cosine.

Compare this series to the series for cosine and hyperbolic sine.

Here are some more useful facts about sine, some gathered together from other nodes, others apparently not yet noded.

The law of sines: In any triangle, the ratio of the sine of an angle to the length of the opposite side is constant. That is,

a / sin A = b / sin B = c / sin C
where we are writing A for the angle opposite side a. Or you can write them as sin A / a. If any of the a or sin A is zero it can't be a triangle, it's just flat.

Exact values of sine, cosine, and tan: There is an easy-to-remember progression of exact values for the three most important acute angles, 30°, 45°, and 60°:

sin 0° = √0 / 2 = 0
sin 30° = √1 / 2 = 0.5
sin 45° = √2 / 2 = 1/2 ≈ 0.7071
sin 60° = √3 / 2 ≈ 0.8660
sin 90° = √4 / 2 = 1
Note this is not simply formulaic: 15° and 75° don't fit in so neatly, but they're less often used. Cosines work in the same way, but downwards from 4 to 0.

Other identities:

sin (−θ) = −sin θ
sin (θ + φ) = sin θ cos φ + cos θ sin φ
sin 2θ = 2 sin θ cos θ
sin2 θ + cos2 θ = 1
sin (θ + 2π) = sin θ
sin θ = cos (&pi/2 − θ)
sin (π − θ) = sin θ

cosecant: The reciprocal of sine is called cosecant, abbreviated cosec or csc. This isn't greatly important as a function in its own right, except as a notational convenience: although the square of sin x is written sin² x, its reciprocal is never written as sin−1 x, that notation being reserved for its inverse function.

arcsine: The inverse of the sine function is arcsine, symbol sin−1 or arsin or arcsin. Since sin is periodic, its inverse is not uniquely defined as a function. Restricting sin to the interval [−π/2, π/2] makes it a one-to-one mapping onto the interval [−1, 1], so we can define a principal arcsine function, symbolized Arcsin or Sin−1. So Arcsin 1/√2 = π/4.

signum: As 'sine' is pronounced the same as 'sign' in English, we have a problem when we actually want to talk about the sign of something: whether it's positive or negative. So the signum function is used, symbol sgn, taking the three values {−1, 0, 1} depending on the sign-with-a-g of its argument. Presumably to be pronounced with the first bit like 'signal'.

Sine (?), n. [LL. sinus a sine, L. sinus bosom, used in translating the Ar. jaib, properly, bosom, but probably read by mistake (the consonants being the same) for an original jiba sine, from Skr. jiva bowstring, chord of an arc, sine.] Trig. (a)

The length of a perpendicular drawn from one extremity of an arc of a circle to the diameter drawn through the other extremity.


The perpendicular itself. See Sine of angle, below.

Artificial sines, logarithms of the natural sines, or logarithmic sines. -- Curve of sines. See Sinusoid. -- Natural sines, the decimals expressing the values of the sines, the radius being unity. -- Sine of an angle, in a circle whose radius is unity, the sine of the arc that measures the angle; in a right-angled triangle, the side opposite the given angle divided by the hypotenuse. See Trigonometrical function, under Function. -- Versed sine, that part of the diameter between the sine and the arc.


© Webster 1913.

Si"ne (?), prep. [L.]



© Webster 1913.

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