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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. 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 θ 1 both x and y are positive; for θ 2 x is negative. 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. Graph of y  = sin θ as a function of the angle θ. The values of sin θ repeat every 360°. 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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