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Cosine: Properties

The cosine function has a number of properties that result from it being periodic and even. 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 cosine function is periodic with a period of 2p, which implies that

cos(q) = cos(q + 2p)
or more generally,
cos(q) = cos(q + 2pk), k Î integers

The function is even; therefore,

cos(-q) = cos(q)

Formula:

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
It is then easily derived that
cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
Or more generally,
cos(x ± y) = cos(x)cos(y) ± sin(x)sin(y)

From the above we can easily derive that

cos(2x) = cos2(x) - sin2(x)
(The cos2(x) is alternate notation for (cos(x))2.)

By observing the graphs of sine and cosine, we can express the cosine function in terms of sine:

cos(x) = sin(x - p/2)

The pythagorean identity gives an alternate expression for cosine in terms of sine

cos2(x) = 1 - sin2(x)

The Law of Cosines relates all three sides and one of the angles of an arbitrary (not necessarily right) triangle:

c2 = a2 + b2 - 2ab cos(C).
where A, B, and C are the angles opposite sides a, b, and c respectively. It can be thought of as a generalized form of the pythagorean theorem.