Definition of cosine

The cosine of an angle is defined as the sine of the complementary angle. The complementary angle equals the given angle subtracted from a right angle, 90°. For instance, if the angle is 30°, then its complement is 60°. Generally, for any angle t,
cos t = sin (90° – t).
Written in terms of radian measurement, this identity becomes
cos t = sin (/2 – t).
Right triangles and cosines

Consider a right triangle ABC with a right angle at C. As mentioned before, we'll generally use the letter a to denote the side opposite angle A, the letter b to denote the side opposite angle B, and the letter c to denote the side opposite angle C. Since the sum of the angles in a triangle equals 180°, and angle C is 90°, that means angles A and B add up to 90°, that is, they are complementary angles. Therefore the cosine of B equals the sine of A. We saw on the last page that sin A was the opposite side over the hypotenuse, that is, a/c. Hence, cos B equals a/c. In other words, the cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse:

cos=adj/hyp.

Also, cos A = sin B = b/c.

The Pythagorean identity for sines and cosines

Recall the Pythagorean theorem for right triangles. It says that
a2 + b2 = c2
where c is the hypotenuse. This translates very easily into a Pythagorean identity for sines and cosines. Divide both sides by c2 and you get
a2/c2 + b2/c2 = 1.
But a2/c2 = (sin A)2, and b2/c2 = (cos A)2. In order to reduce the number of parentheses that have to be written, it is a convention that the notation sin2 A is an abbreviation for (sin A)2, and similarly for powers of the other trig functions. Thus, we have proven that
sin2 A + cos2 A = 1
when A is an acute angle. We haven't yet seen what sines and cosines of other angles should be, but when we do, we'll have for any angle t one of most important trigonometric identities, the Pythagorean identity for sines and cosines:
sin2 t + cos2 t = 1.