cosine

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trigonometry

trigonometry [Gr.,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in surveying, navigation, and astronomy.

The Basic Trigonometric Functions

Trigomometry originated as the study of certain mathematical relations originally defined in terms of the angles and sides of a right triangle, i.e., one containing a right angle (90°). Six basic relations, or trigonometric functions, are defined. If A, B, and C are the measures of the angles of a right triangle (C=90°) and a, b, and c are the lengths of the respective sides opposite these angles, then the six functions are expressed for one of the acute angles, say A, as various ratios of the opposite side (a), the adjacent side (b), and the hypotenuse (c), as set out in the table. Although the actual lengths of the sides of a right triangle may have any values, the ratios of the lengths will be the same for all similar right triangles, large or small; these ratios depend only on the angles and not on the actual lengths. The functions occur in pairs—sine and cosine, tangent and cotangent, secant and cosecant—called cofunctions. In equations they are usually represented as sin, cos, tan, cot, sec, and csc. Since in ordinary (Euclidean) plane geometry the sum of the angles of a triangle is 180°, angles A and B must add up to 90° and therefore are complementary angles. From the definitions of the functions, it may be seen that sin B=cos A, cos B=sin A, tan B=cot A, and sec B=csc A; in general, the function of an angle is equal to the cofunction of its complement. Since the hypotenuse (c), is always the longest side of a right triangle, the values of the sine and cosine are always between zero and one, the values of the secant and cosecant are always equal to or greater than one, and the values of the tangent and cotangent are unbounded, increasing from zero without limit.

For certain special right triangles the values of the functions may be calculated easily; e.g., in a right triangle whose acute angles are 30° and 60° the sides are in the ratio 1 : √3  : 2, so that sin 30°=cos 60°=1/2, cos 30°=sin 60°=√3/2, tan 30°=cot 60°=1/√3, cot 30°=tan 60°=√3, sec 30°=csc 60°=2/√3, and csc 30°=sec 60°=2. For other angles, the values of the trigonometric functions are usually found from a set of tables or a scientific calculator. For the limiting values of 0° and 90°, the length of one side of the triangle approaches zero while the other approaches that of the hypotenuse, resulting in the values sin 0°=cos 90°=0, cos 0°=sin 90°=1, tan 0°=cot 90°=0, and sec 0°=csc 90°=1; since division by zero is undefined, cot 0°, tan 90°, csc 0°, and sec 90° are all undefined, having infinitely large values.

A general triangle, not necessarily containing a right angle, can also be analyzed by means of trigonometry, and various relationships are found to exist between the sides and angles of the general triangle. For example, in any plane triangle a/sin A=b/sin B=c/sin C. This relationship is known as the Law of Sines. The related Law of Cosines holds that a2=b2+c2−2bc cosA and the Law of Tangents holds that (ab)/(a+b)=[tan 1-2(AB)]/[tan 1-2(A+B)]. Each of the trigonometric functions can be represented by an infinite series.

Extension of the Trigonometric Functions

The notion of the trigonometric functions can be extended beyond 90° by defining the functions with respect to Cartesian coordinates. Let r be a line of unit length from the origin to the point P (x,y), and let θ be the angle r makes with the positive x-axis. The six functions become sin θ =y/r=y, cos θ=x/r=x, tan θ=y/x, cot θ=x/y, sec θ=r/x=1/x, and csc θ=r/y=1/y. As θ increases beyond 90°, the point P crosses the y-axis and x becomes negative; in quadrant II the functions are negative except for sin θ and csc θ. Beyond θ=180°, P is in quadrant III, y is also negative, and only tan θ and cot θ are positive, while beyond θ=270° P moves into quadrant IV, x becomes positive again, and cos θ and sec θ are positive. Since the positions of r for angles of 360° or more coincide with those already taken by r as θ increased from 0°, the values of the functions repeat those taken between 0° and 360° for angles greater than 360°, repeating again after 720°, and so on. This repeating, or periodic, nature of the trigonometric functions leads to important applications in the study of such periodic phenomena as light and electricity.

cosine

of an angle
a trigonometric function that in a right-angled triangle is the ratio of the length of the adjacent side to that of the hypotenuse; the sine of the complement
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

COSINE

Cooperation for Open Systems Interconnection Networking in Europe. A EUREKA project.

sine

In a right triangle, the ratio of the side opposite an acute angle (less than 90 degrees) and the hypotenuse. The cosine is the ratio between the adjacent side and the hypotenuse. These angular functions are used to compute circular movements.
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References in periodicals archive ?
Applying IEEE standard 1547-2003 [20], for complete analysis of the signal in distributive network, 50 harmonics has to be measured, where every harmonic has it's cosine and sine component.
So with the acquisition of cosine similarity, the results can be categorized of the similarity from each word sent by the perpetrator and then implemented into the system to assist investigators in the investigation of a case of conversation, especially on text messages.
(i) In this paper, we propose sine, cosine and cotangent similarity measures under interval rough neutrosophic environment.
The amplitudes of cosine harmonics are much smaller than their periods ([[DELTA].sub.i] [much less than] [[lambda].sub.i]; where i = 1, 2...
Next, we formulate the following properties for Riemann-Liouville derivatives of the hyperbolic sine and hyperbolic cosine functions.
Let A = ([[mu].sub.A]([X.sub.j]), [v.sub.A]([X.sub.j])) and B = ([[mu].sub.B]([X.sub.j]), [v.sub.b]([X.sub.j])) be two IFSs in X, the cosine similarity measure between A and B is defined as follows:
Liu, "Parameter optimization of support vector regression based on sine cosine algorithm," Expert Systems with Applications, vol.
A cosine operator function [T.sub.h] [member of] [X] (h [greater than or equal to] 0) is defined by the properties
First, we briefly recall some definitions from the theory of cosine family [5, pp.
DCT can map an original data into frequency domain by cosine waveform, and conversely inverse discrete cosine transform (IDCT) transfers frequency domain data into spatial domain.
The aim of this paper is to introduce the interval-valued intuitionistic fuzzy ordered weighted cosine similarity (IVIFOWCS) measure.

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