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'.