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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.
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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Sine

 

(sin), a trigonometric function. The sine of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The Indian mathematicians used the word jiva, whose literal meaning is “bowstring,” for the sine. The Arabs rendered jiva as jiba and subsequently used the reading jaib, which is a common word in Arabic meaning “curve,” “bosom,” or “fold” and corresponding to the Latin word sinus.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.

sine

[sīn]
(mathematics)
The sine of an angle A in a right triangle with hypotenuse of length c given by the ratio a / c, where a is the length of the side opposite A ; more generally, the sine function assigns to any real number A the ordinate of the point on the unit circle obtained by moving from (1,0) counterclockwise A units along the circle, or clockwise | A | units if A is less than 0. Denoted sin A.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

sine

of an angle
a. a trigonometric function that in a right-angled triangle is the ratio of the length of the opposite side to that of the hypotenuse
b. a function that in a circle centred at the origin of a Cartesian coordinate system is the ratio of the ordinate of a point on the circumference to the radius of the circle
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

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.
Copyright © 1981-2019 by The Computer Language Company Inc. All Rights reserved. THIS DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher.
References in periodicals archive ?
For SINEs and LINEs, correlation coefficients between all pair-wise species were strongly positive (Table 1).
On Saturday, the juggling continues with plans for a rare "above the nameplate" King-related headline on Page One, with Final Four coverage below the name, Sines said.
In contrast to standard sequencing, analysis of SINEs and LINEs presents a nearly perfect record of evolutionary change, says Okada.
What sparked Sines' interest in doing a statewide series was his appointment by the governor to a special youth task force to examine the issue.
Labour's Birmingham MP Jack Dromey agreed Sines, who runs 16 sites with partner James Crickmore, "has a long track record of abusing those who live on park homes sites that he owns".
A buyer offered pounds 48,000 in March and paid a deposit but dropped out after speaking to Sines and offered no explanation.
Sines and Crickmore received the heftiest bans from the BHA for their role as ringleaders in a case that was described as unprecedented in its scale and complexity by BHA security director Paul Scotney.
Sines admitted telling one on Medina Park their home was "junk" and "worthless" with the aim of getting them to give up the property.
In reasons for punishments published on Monday, it emerged that an application was made on Crickmore's behalf that he should receive a shorter sentence than the 14 years handed to co-conspirator Maurice Sines because "racing played a more important part in his social life".
The resulting exclusion orders bring to 78.5 years the total bans arising from the investigation and subsequent 11-day inquiry from which the longest punishments, of 14 years each, were handed down to former owners Maurice Sines and James Crickmore.