# Math2.org Math Tables: (Math)
 Sine Topics overview formal definition graph properties expansions derivative integral Sine: Properties The sine function has a number of properties that result from it being periodic and odd. Most of these should not be memorized by the reader; yet, the reader should be able to instantly derive them from an understanding of the function's characteristics. The sine function is periodic with a period of 2p, which implies that sin(q) = sin(q + 2p) or more generally, sin(q) = sin(q + 2pk), k ö integers The function is odd; therefore, sin(-q) = -sin(q) Formula: sin(x + y) = sin(x)cos(y) + cos(x)sin(y) It is then easily derived that sin(x - y) = sin(x)cos(y) - cos(x)sin(y) Or more generally, sin(x Ý y) = sin(x)cos(y) Ý cos(x)sin(y) From the above we can easily derive that sin(2x) = 2sin(x)cos(x) By observing the graphs of sine and cosine, we can express the sine function in terms of cosine: sin(x) = cos(x - p/2) The pythagorean identity gives an alternate expression for sine in terms of cosine sin2(x) = 1 - cos2(x) The Law of Sines relates various sides and angles of an arbitrary (not necessarily right) triangle: sin(A)/a = sin(B)/b = sin(C)/c. where A, B, and C are the angles opposite sides a, b, and c respectively.