Cosine
Cosine is a trigonometric function that provides information about the dimensions of an angle using a ratio. An angle can be illustrated geometrically as a slice of a circle. Consider angle AOB positioned within a unit circle centered at the origin (0,0) of the Cartesian coordinate system. A unit circle is defined by x2+y2=1, where 1 is the radius. The vertex of the angle (i.e., the point at which the legs intersect) is at the origin. The initial leg of the angle lies upon the positive x-axis and intersects the circle at point A with coordinates (1,0). The terminal leg intersects the circle at point B, which has the coordinates (x,y).
By definition, the cosine of angle AOB is cos AOB=x/r. This is true for any circle described by x2+y2=r2. Because r=1 in this particular case, cos AOB=x (which is the distance from O to x and is sometimes called the abscissa).
The angle is said to be subtended by arc AB. The length of arc AB is a portion of the entire circle circumference 2r. The angle AOB is the same portion of the entire circle. Angle measurements are expressed in radians or degrees (i.e., sexagesimal numeration). A radian is defined as the angle size for which the subtended arc is equal in length to the radius of the circle. A degree is defined as 1/360 of a circle, because there are 360° in one complete revolution. The two measures are approximately related by 1 radian=57.29578° and 1°=0.01745 radian.
Assume that angle AOB is ⅛ of the circle. Because a circle is 2 radians, angle AOB is ⅛x2=¼ radians or 0.7854 radians. Likewise, it is ⅛ of 360° or 45°. The cosine of a 45° angle to six decimal places is 0.707107. This value is available from trigonometric tables or can be calculated with a calculator.
If an angle is a real number expressed in radians, its cosine can be approximated using the first few terms of the following infinite series:
cos =1-(2/2!)+(4/4!)-(6/6!)+(8/8!) - etc. Recall that a factorial is the product of all positive integers leading up to and including the indicated integer (e.g., 8!=1x2x3x4x5x6x7x8=40,320).
Therefore, the cosine of 45° (¼ or 0.7854 radians), can be approximated as follows: cos 0.7854=1-(0.6169/2)+(0.3805/24)-(0.2347/720)+(0.1448/40,320)=0.707082. This is a very good approximation.
Angle AOB is considered a positive angle, because its terminal leg rotates in a counterclockwise direction from its initial arm. If the terminal leg rotated in a clockwise direction, the angle would be negative. The cosine of any angle is always between -1 and 1. The cosine is positive for angles terminating in the first and fourth quadrants and negative for angles terminating in the second and third quadrants. For example, an angle of -100° terminates in the third quadrant, therefore its cosine is negative. However, the cosine of a negative angle is equivalent to the cosine of its positive value (e.g., cos -100°=cos 100°).
Because cosine is a periodic (repeating) function with a period of 2, the cosine of any angle equals the cosine of the sum of that angle with any multiple of 360°. In other words, the terminal leg could be rotated many more times around the circle, and as long as it returns to the same position relative to the initial leg, the cosine of the angle would still be the same. Thus, cos 72°=cos (72+360°)=cos 432°.
The cosine and sine of an angle are closely related. By definition, the sine of example angle AOB is sin AOB=y/r. Because r=1 in this case, sin AOB=y (which is the distance from O to y and is sometimes called the ordinate). The sine and cosine of any angle are related by the following equation: cos =sin (90°-) in degrees and cos =sin (½-) in radians. This equation is true even if the angle size exceeds 360° (2 radians) or if the angle is negative.
Cosine, and the other trigonometric functions, are very useful for solving triangles (i.e., finding a leg length or angle given enough information about other leg lengths and/or angles). Consider the triangles below comprised of three legs (a, b, and c) and containing three angles (A, B, and C). The triangle on the left is called an oblique triangle, because all of its angles are acute (i.e., less than 90°). The other triangle is a right triangle, because it contains a right angle of 90°.
The law of cosines states the following for any triangle: (The square of any one leg length)=(The sum of the squares of the other 2 leg lengths)-(Their product times the cosine of the angle between them).
- a2=(b2+c2)-(2bc cos A)
- b2=(a2+c2)-(2ac cos B)
- c2=(a2+b2)-(2ab cos C).
For the special case of the right triangle, leg c is the hypotenuse and angle C is 90°. Because cos 90°=0, the equation for leg c reduces to the Pythagorean Theorem: c2= a2+b2.
In a right triangle, the cosine of each acute angle equals the length of the adjacent side divided by the length of the hypotenuse (i.e., cos A=b/c and cos B=a/c). This is a very useful formula that can be proven using the equation given earlier to describe angle AOB in a unit circle. Think of leg length c as the radius of the circle and leg length b as the x value. Then, cos AOB=x/r is equivalent to cos A=b/c.
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