What is Trigonometry?

Trigonometry is a branch of mathematics that developed from simple mensuration (measurement of geometric quantities), geometry, and surveying. In its modern form it makes use of concepts from algebra and analysis. Initially it involved the mathematics of practical problems, such as construction and land measurement; it has since been extended to the geometry of three-dimensional spaces in the form of SPHERICAL TRIGONOMETRY. This article, however, will deal only with plane trigonometry.

Land survey makes use of the process of triangulation, in which a chosen network of triangles is measured. In pure triangulation, a base line to one triangle is measured, and the rest of the survey involves measuring angles only; in mixed triangulation, certain sides and angles are measured; in chain triangulation, only sides are measured.

Basic Concepts

Trigonometric concepts are used to minimize the amount of measuring involved. These concepts depend on the concepts of enlargement and similarity. Equiangular triangles have the same shape, but only in the special case of congruency do they have the same size. Any set of similar triangles has the invariant property of proportionality; that is, ratios of pairs of corresponding sides are in the same proportion. In the language of transformation geometry, for similar triangles, one triangle is an enlargement of another, or any triangle can be transformed into another by applying the same scale factor to each part of the triangle. In the case of a fractional scale factor the enlargement is, in fact, a reduction.

Ratios

Trigonometry uses the fact that ratios of pairs of sides of triangles are functions of the angles. The basis for mensuration of triangles is the right-angled triangle. The term trigonometry means literally the measurement of trigons (triangles). This mensuration approach defines the six trigonometric ratios in terms of ratios of lengths of sides of a right triangle. The analytic approach defines the ratios in terms of the coordinates of a point on the circumference of a unit circle, xx + yy = 1. These ratios define the trigonometric functions.

If A is an acute angle, the trigonometric ratios of A are conveniently defined as ratios of different lengths of the corresponding right triangle: the hypotenuse, the side adjacent to the angle A, and the side opposite to A. 

             sine (sin) A = opp/hyp 
             
             tangent (tan) A = opp/adj 
             
             secant (sec) A = hyp/adj 
             
             cosine (cos) A = adj/hyp 
             
             cotangent (cot, ctn) A = adj/opp 
             
             cosecant (cosec, csc) A = hyp/opp
The coratios are the ratios of complementary angles (angles whose sum is 90 deg); for example, cos A = sin (90 deg - A). Every ratio has a reciprocal ratio.

Trigonometric Functions

Trigonometric functions, often known as the circular functions, are defined in terms of the trigonometric ratios. If point P (x, y) lies on the circumference of a unit circle xx + yy = 1, then the trigonometric functions of A, where A is the angle that the line OP makes with the positive direction of the x-axis, are defined as: 
            
             sin A = y 
             
             tan A = y/x 
             
             sec A = 1/x 
             
             cos A = x 
             
             cot A = x/y 
             
             csc A = 1/y
The signs of the coordinates determine the signs of the ratios. If A is acute, all ratios are positive; these values of A are fully tabulated in standard trigonometric tables. The ratios of angles greater than a right angle (90 deg) can be converted to ratios of acute angles by appropriate reduction formulas. The reduction formulas are trigonometric identities that express the trigonometric ratios of an angle of any size in terms of the trigonometric ratios of an acute angle.

If y = sin x, then the inverse statement, that x is the angle whose sine is y, is written x = inverse sin y, or arc sin y. Trigonometric functions have many applications in algebra. They are used in rationalizing quadratic surds (square roots). For example, the algebraic function y = the square root of (aa + xx) can be transformed into the rational trigonometric function y = a sec u using the identity 1 + tan u tan u = sec u sec u and substitution x = a tan u. Similarly, y = the square root of (aa - xx) and y = the square root of (nn - aa) can be rationalized by suitable use of identities and substitutions. Substitutions have various uses in facilitating processes in the CALCULUS.

Trigonometric functions also have value in applied mathematics. For example, all oscillations can be represented as PERIODIC FUNCTIONS, that is, functions that repeat their values at equal intervals of the independent variable. Any periodic function can be represented by an infinite trigonometric series. A Fourier series is an example (see FOURIER ANALYSIS).

Polar Coordinates

Trigonometric functions are used in polar coordinates, the system in which the position of a point P is determined by its distance OP from a fixed point O and by the angle that OP makes with an initial line OX (see COORDINATE SYSTEMS, mathematics). The analytic definition of trigonometric functions above uses the special case of polar coordinates when the distance OP is unity. In the general application to points in a plane, point O is the pole; OP is the radius vector of P; OX is the polar axis; angle XOP or A, measured counterclockwise, is called the polar angle, vectorial angle, azimuth, or amplitude of P. The Cartesian coordinates (x,y) of P, when O is the origin and OX is the x-axis, are related by the equations x = r cos A, y = r sin A and r = the square root of (xx + yy), A = inverse tan (y/x). This system can be extended to form spherical coordinates in space.

Identity

If a trigonometric equation is true for all values of its variables, it is an identity. Some trigonometric identities state relations between various combinations of the six trigonometric functions determined by their definitions. Others are trigonometric forms of classical geometric theorems.

The theorem of Pythagoras, aa = bb + cc, for a right triangle ABC with angle A being the right angle, can be transformed by replacing b and c by a sin B and a cos B, respectively. The simplified identity is then, for any angle M, sin M sin M + cos M cos M = 1. By using the identity tan M = sin M/cos M, alternative forms can be derived: tan M tan M + 1 = sec M sec M and cot M cot M + 1 = csc M csc M. The extension of the theorem of Pythagoras can be written as the cosine formula (cosine rule) aa = bb + cc - 2bc cos A, with similar versions for cos B and cos C that are obtained by changing the letters in cyclic order.

The six trigonometric ratios for single angles can be used to form the addition and subtraction formulas. 

             
             sin (A + B) = sin A cos B + cos A sin B 
             
             sin (A - B) = sin A cos B - cos A sin B 
             
             cos (A + B) = cos A cos B - sin A sin B 
             
             cos (A - B) = cos A cos B + sin A sin B 
             
             tan (A + B) = (tan A + tan B)/(1 - tan A tan B) 
             
             tan (A - B) = (tan A - tan B)/(1 + tan A tan B)
If B is set equal to A in the above addition formulas, the following double-angle formulas are obtained. 

             
             sin 2A = 2 sin A cos A 
             
             cos 2A = cos A cos A - sin A sin A 
             
             tan 2A = (2 tan A)/(1 - tan A tan A)
By setting B = 2A, triple-angle formulas can be obtained for sin 3A, cos 3A, and tan 3A. The sine formula (sine rule), which applies to any triangle ABC with sides a, b, and c, is a/sin A = b/sin B = c/sin C = 2R where R is the radius of the circumscribing circle. These formulas are useful in the solution of triangles, that is, the determination of measurements of a triangle from given data. The reduction formulas in trigonometry are identities that express trigonometric ratios of any angle in terms of ratios of acute angles. For example, cos (180 deg + A) = -cos A, where A is an acute angle.

Apart from their use in simplifying trigonometric problems, trigonometric identities are often used in algebra and CALCULUS.

Alaric Millington

Bibliography: Drooyan, Irving, et al., Essentials of Trigonometry, 2d ed. (1977); Heineman, E. Richard, Plane Trigonometry with Tables, 4th ed. (1974); Keedy, Mervin L., and Bittinger, Marvin L., Trigonometry: Triangles and Functions, 2d ed. (1978); Swokowski, Earl W., Fundamentals of Trigonometry, 4th ed. (1978); Wooton, William, et al., Modern Trigonometry, rev. ed. (1976).