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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 thatsin(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.
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