trigonometry: 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 :
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 a 2= b 2+ c 2−2 bc cos A and the Law of Tangents holds that ( a − b )/( a + b )=[tan 1⁄2 ( A − B )]/[tan 1⁄2 ( A + B )]. Each of the trigonometric functions can be represented by an infinite series .
The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.
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