describing angle measures. When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2 (≈ 6.28) rad. For real number x, the notations sin x, cos x, etc. refer to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/)°, so that, for example, sin = sin 180° when we take x = . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = /180 ≈ 0.0175. 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 of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between and radians the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers. Let be the ray obtained by rotating by an angle the positive half of the -axis (counterclockwise rotation for and clockwise rotation for ). This ray intersects the unit circle at the point The ray extended to a line if necessary, intersects the line of equation at point and the line of equation at point The tangent line to the unit circle at the point , is perpendicular to and intersects the - and -axes at points and The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of in the following manner. The trigonometric functions and are defined, respectively, as the x- and y-coordinate values of point . That is, and In the range , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius as hypotenuse. And since the equation holds for all points on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity. The other trigonometric functions can