# 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.;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 θ_{1} 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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