Sine and Cosine
Sine and Cosine: Properties
The sine function has a number of properties that result from it being periodic and odd. The cosine function has a number of properties that result from it being periodic and even. Most of the following equations 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 and cosine functions are periodic
with a period of 2p. This implies that
sin(q) = sin(q + 2p)
cos(q) = cos(q + 2p)or more generally,
sin(q) = sin(q + 2pk)
cos(q) = cos(q + 2pk),where k Î integers.
The sine function is odd; therefore,
sin(-q) = -sin(q)
The cosine function is even; therefore,
cos(-q) = cos(q)
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)
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 sine equation, we can derive that
sin(2x) = 2sin(x)cos(x)From the above cosine equation, we can derive that
cos(2x) = cos2(x) - sin2(x)(The notation sin2(x) is equivalent to (sin(x))2. Warning: sin-1(x) stands for arcsin(x) not the multiplicative inverse of sin(x).)
By observing the graphs of sine and cosine, we can express
the sine function in terms of cosine and vice versa:
sin(x) = cos(90° - x)and the cosine function in terms of sine:
cos(x) = sin(90° - x)Such a trig function (f) that has the property
f(q) = g(complement(q))is called a cofunction of the function g, hence the names "sine" and "cosine."
The pythagorean identity,
sin2(x) = 1 - cos2(x)
cos2(x) = 1 - sin2(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 = 2r.where A, B, and C are the angles opposite sides a, b, and c respectively. Furthermore, r is the radius of the circle circumscribed in that triangle.
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. Warning: You must be careful when solving for one of the sides adjacent to the angle of interest, for there will often be two triangles that satisfy the given conditions. This can be understood from geometry. A triangle defined by SAS (side-angle-side) is unique, and, therefore, any triangle with the same SAS parameters must be congruent to it. A triangle defined by SSA, however, is not always unique, and two triangles with the same SSA parameters may or may not be congruent.