Trigonometric functions
Unit-circle definitions

The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.

Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis (counterclockwise rotation for θ > 0, {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0}). This ray intersects the unit circle at the point A = (x A, y A). {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L, {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = (1, y B), {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and the line of equation y = 1 {\displaystyle y=1} at point C = (x C, 1). {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A, is perpendicular to L, {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = (0, y D) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = (x E, 0). {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.

The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A. That is,

In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2}, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse. And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = (x, y) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity.

The other trigonometric functions can be found along the unit circle as

By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is

Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi }. Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi }. That is, the equalities

hold for any angle θ and any integer k. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after a rotation by an angle π {\displaystyle \pi }, the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi }. That is, the equalities

hold for any angle θ and any integer k.
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