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length and arc

TANGENT, a straight line of indefinite length, which touches but does not cut a curve; also the length of a straight line which touches a curve measured from the point of tangency to the point where it meets a diameter of the curve; one of the trigonometrical functions. Tan gent to a curve is the limiting position of a secant. Suppose a straight line as cutting a curve in two points near to one another, and then suppose the line to move so that the points approach each other; at the instant when the points coincide the line is a tangent to the curve. Let A a be any arc less than 90, draw A Et touch ing the arc at A; from the cen tre c draw c s a, cutting A I; in a ; the length A x is the tangent of the arc A a. It is now considered best to make a 1 distinction between the tangent of an arc and the tangent of an angle. An arc is a curved line of certain length; an angle is not measured by the length of arc, for the measure of a certain angle is so many degrees, whatever may be the length in inches of the circular arc subtending it. The

tangent of an angle is called a trigonometrical ratio, because it is the fraction formed by di viding the number representing the length of one side of a right-angled triangle by that repre senting the length of the other side. Consider the triangle H c A, the tangent of the angle A II H C A iS -; this fraction is the same whatever CA the lengths of A H and c-A. (See TruGoNoM my). A plane is said to be tangent to a curved surface when three points of the plane coin cide with three points very close together of the surface. A list of the properties involving tangents may be obtained from works on analytical geometry.