[justify]On the unit circle, we define the sine of an angle [math]\theta[/math], denoted [math] \sin \theta[/math], as the [math]y[/math]-coordinate of the terminal point of [math] \theta[/math] and the cosine of  [math] \theta[/math], denoted [math] \cos \theta[/math] as the [math]x[/math]-coordinate of the terminal point of [math] \theta[/math].  Recall that the unit circle is the graph of the equation [math] x^2+y^2=1[/math]. By substituting [math] x = \cos \theta[/math] and [math] y = \sin \theta,[/math] we immediately have the Pythagorean identity[/justify][center][math]\Large \cos^2 \theta + \sin^2 \theta = 1.[/math][/center]Since the arc length is the same as the angle subtended on the unit circle, [math] \theta[/math] can be considered an angle or a real number corresponding to the arc length on the unit circle starting from the point [math](1,\,0)[/math].  Counter-clockwise rotation is considered a positive direction of rotation, and clockwise is negative.  

[justify]In circles of arbitrary radius [math]r,[/math] the arc length, [math]x,[/math] and [math]y[/math] coordinates all grow or shrink proportionally to [i][math]r.[/math]  [/i]So the coordinates of the terminal point [i][math]P[/math] [/i]are [math] \large x = r \cos \theta[/math] and [math] \large y = r \sin \theta[/math].   Dividing both sides of both equations by [math]r[/math] gives the more general definitions for sine and cosine for a circle of any radius.[/justify][center][math]\Large \cos \theta = \frac{x}{r}, \ \ \sin \theta = \frac{y}{r}[/math][/center][br]

[justify]The circle of radius [i][math]r[/math][/i] centered at the origin is defined by the equation [math] x^2 + y^2 = r^2[/math]. Substituting for [i][math]x[/math][/i] and [i][math]y[/math][/i] we have [math] (r \cos \theta)^2 + (r \sin \theta )^2 = r^2[/math]. Dividing both sides by [math] r^2[/math], we see that the Pythagorean identity [math] \cos^2 \theta + \sin^2 \theta = 1[/math] holds for a circle of any radius.  [/justify]