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 2`θ` = 2 sin `θ` cos `θ`

sin^{2} `θ` + cos^{2} `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'.