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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 Î 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 ± y) = sin(x)cos(y) ± 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.