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.

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/oppThe 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.

sin A = y tan A = y/x sec A = 1/x cos A = x cot A = x/y csc A = 1/yThe 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).

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.