Tangent (trigonometry).

Tangent (trigonometry). Tangent (trigonometry), one of the six fundamental ratios of trigonometry. The others are sine, cosine, secant, cosecant, and cotangent. A ratio is a proportional relationship between two numbers calculated by dividing one number by the other. Tangent embodies such a relationship between the magnitudes of the angles of a right triangle (a triangle having one 90° angle) and the lengths of the sides. Varying one value, such as the magnitude of an angle, requires the related value, such as the length of a side, to change in a predictable way. The tangent, usually abbreviated tan, of one of the acute (less than 90°) angles of a right triangle is found by dividing the length of the side opposite the angle by the length of the shorter of the two sides adjacent to the angle. (The other adjacent side is called the hypotenuse, and is the triangle's longest side.) If the acute angle in question is called theta (?), then . Tangent smoothly increases in numerical value from 0 to infinity as the angle increases from 0° to 90°. Tangent is also defined for angles greater than 90° using right triangles inscribed in a circle centered at the point (0,0) on the xy axis: A line drawn from the circle's center to any point on the circle makes an angle, ? , with the x axis. The tangent of ? is equal to the vertical distance of the point from the x axis divided by the horizontal distance of the point from the y axis. At 90°, tangent is discontinuous, flipping from positive infinity to negative infinity. The function rises from negative infinity to 0 at 180° and continues to climb, approaching positive infinity at 270°. At 270° tangent is again discontinuous, flipping from positive to negative infinity. Beyond 270° tangent increases in numerical value, reaching 0 at 360°. Cotangent is tangent's reciprocal function. The cotangent, usually abbreviated cot, of an acute angle of a right triangle is equal to the length of the shorter side adjacent to the chosen acute angle divided by the length of the side opposite the angle: . Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.