# trigonometry: Extension of the Trigonometric Functions

The notion of the trigonometric functions can be extended beyond 90° by defining the functions with respect to Cartesian coordinates . Let * r * be a line of unit length from the origin to the point * P * ( * x,y *), and let θ be the angle * r * makes with the positive * x *-axis. The six functions become sin θ = * y */ * r *= * y, * cos θ= * x */ * r *= * x, * tan θ= * y */ * x, * cot θ= * x */ * y, * sec θ= * r */ * x *=1/ * x, * and csc θ= * r */ * y *=1/ * y. * As θ increases beyond 90°, the point * P * crosses the * y *-axis and * x * becomes negative; in quadrant II the functions are negative except for sin θ and csc θ. Beyond θ=180°, * P * is in quadrant III, * y * is also negative, and only tan θ and cot θ are positive, while beyond θ=270° * P * moves into quadrant IV, * x * becomes positive again, and cos θ and sec θ are positive. Since the positions of * r * for angles of 360° or more coincide with those already taken by * r * as θ increased from 0°, the values of the functions repeat those taken between 0° and 360° for angles greater than 360°, repeating again after 720°, and so on. This repeating, or periodic, nature of the trigonometric functions leads to important applications in the study of such periodic phenomena as light and electricity.

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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