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.;g639;none;1;g639;;;block;;;;no;1;4224n;65328n;;;;;trigonome639;;;left;stack;;;;;;;;left;stack;2745n;;;The trigonometric functions of the angle formed by the x -axis and the line r terminating at point P may be expressed in terms of r and the x- and y- coordinates of P. For θ1 both x and y are positive; for θ2 x is negative.CE5Since 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.;g640;none;1;g640;;;block;;;;no;1;4224n;173883n;;;;;trig-sin640;;;left;stack;;;;;;;;left;stack;2745n;;;Graph of y   =  sin θ as a function of the angle θ. The values of sin θ repeat every 360°.CE5This repeating, or periodic, nature of the trigonometric functions leads to important applications in the study of such periodic phenomena as light and electricity.

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

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