Cotangent.

Cotangent. Cotangent, one of the six fundamental ratios of trigonometry, along with tangent, sine, cosine, secant, and cosecant. (A ratio is a proportional relationship between two numbers calculated by dividing one number by the other.) Cotangent embodies the relationship between the magnitudes of the angles of a right triangle--a triangle with one 90° angle--and the lengths of its sides. Varying one of the numbers, such as the length of a side, requires the related number, such as the magnitude of an angle, to change in a predictable way. The cotangent, usually abbreviated cot, of one of the acute (less than 90°) angles of a right triangle is equal to the length of the shorter of the two sides adjacent to the angle divided by the length of the side opposite the angle: . (The longer adjacent side is called the triangle's hypotenuse). Cotangent smoothly decreases in numerical value from infinity to 0 as the angle increases from 0° to 90°. The function 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 cotangent of ? is equal to the horizontal distance of the point from the y axis divided by the vertical distance of the point from the x axis. Cotangent smoothly decreases in numerical value from 0 to negative infinity as ? increases from 90° to 180°. At 180°, cotangent is discontinuous, flipping from negative infinity to positive infinity. The function falls from positive infinity to 0 at 270° and continues to decline, approaching negative infinity at 360°. Tangent (abbreviated tan) is cotangent's reciprocal function. The tangent of an acute angle of a right triangle is equal to the length of the side opposite the chosen acute angle divided by the length of the shorter of the two sides adjacent to the angle: Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved. .