trigonometry
trigonometry
[‚trig·ə′näm·ə·trē]Trigonometry
the branch of mathematics in which trigonometric functions and their applications to geometry are studied. Trigonometry is divided into plane trigonometry and spherical trigonometry. The theory of trigonometric functions and its applications to the solution of plane right and oblique triangles are studied in secondary school.
Consider a plane triangle with sides a, b, and c and opposite angles A, B, and C. It can be shown that A + B + C = π. Suppose the altitudes of the triangle are ha, hb, and hc; the perimeter is 2p; the area is S; and the diameter of a circle circumscribed about the triangle is 2R. The following are fundamental relations of plane trigonometry:
(1) Law of sines:
(2) Law of cosines:
a2 = b2 + c2 –2bc cos A
(3) Law of tangents:
(4) Formula for the area of a triangle:
If the sides of a triangle are known, the angles can be found from the law of cosines or from equations of the form
Although the development of plane trigonometry began later than that of spherical trigonometry, some of the theorems of plane trigonometry were known in earlier times. For example, the 12th and 13th theorems of the second book of Euclid’s Elements (third century B.C.) essentially express the law of cosines. Important contributions to the development of plane trigonometry were made by al-Battani (second half of the ninth and early tenth centuries), Abu al-Wafa (tenth century), Bhaskara (12th century), and Nasir al-Din al-Tusi (13th century), who were aware of the law of sines. The law of tangents was derived by Re-giomontanus (15th century). Further contributions were made by Copernicus (first half of the 16th century), T. Brahe (second half of the 16th century), F. Vieta (16th century), and J. Kepler (late 16th and first half of the 17th centuries). Trigonometry assumed its modern form in the work of L. Euler (18th century).