"Stargazers" introduced two ways of describing the position of a point P on a flat plane (e.g. a sheet of paper): cartesian coordinates (x,y) and polar coordinates (r,f). Both used for reference a point O ("origin") and some straight line through it ("xaxis"). In cartesian coordinates a second "yaxis" is drawn through O, perpendicular to the first, and lines parallel to the axes are then dropped from P, cutting the axes at the points A and B on the drawing. The distances OA and OB then give the two numbers which define P, the x and y coordinates of the point. In polar coordinates, the point P is defined by its distance r from the origin O (see drawing) and by its polar angle ("azimuth" on a map) between the xaxis and the "radius" r = OA, measured counterclockwise. Since the figure OAPB is a rectangle, the distance AP also equals y. Therefore sinf = y/r Multiplying everything by r gives the relation between the two systems of coordinates (symbols standing next to each other are understood to be multiplied): x = r cosf These relations allow (x,y) to be calculated when (r,f) are given. To go in the opposite directiongiven (x,y), find (r,f)one notes that in the triangle OAP, by Pythagoras x^{2} + y^{2} = r^{2} Therefore, given (x,y), r can be calculated, and then (sinf, cosf) can be derived as before by sinf = y/r (except at the origin point O, where (x, y, r) are all zero and the above fractions become 0/0; any value can then be chosen for the angle f). However, there remains a problem. The angle f as defined above can go from 0 to 360°, but (sinf, cosf) are only defined for 0 to 90°, covering only the part of the plane where both x and y are positive. When one or both are negative, the angle f is larger than 90 degrees, and such angles never appears in any rightangled triangle. What sort of meaning can (sinf, cosf) have for f larger than 90 degrees? There is a simple solution, though: use the above equations to redefine sinf and cosf for such larger angles! The equations are sinf = y/r They are now viewed as new definitions of the sine and cosine, for the polar angle f given by x and y. If (x,y) are both positive, the result is exactly the same as for angles inside a rightangled triangle. But it also works for larger angles. The sine and cosine can now be negative (just like x and y) but their magnitude still cannot exceed 1, because the magnitude of x and y is never larger than r. Here are those signs: 
Range  sinf = y/r  cosf = x/r 

090°  +  + 
90° 180°  +   
180°  270°     
270°360°    + 
Allowing the line OP to go around the origin more than once allows the angle f to grow past 360°; the sine and cosine are still defined as y/r and x/r, and repeat their previous values. Similarly, turning OA in the opposite directionclockwisecan define negative values of f. Together, these extensions define (sinf, cosf) for any angle f, positive or negative, of any size. The relation derived from Pythagoras' theorem sin^{2}f + cos^{2}f = 1 holds for any of those angles. If either the sine or the cosine is zero, the other function must be +1 or 1, depending on the sign of the coordinate (x or y) that defines them. At 90° and 270°, x = 0 and therefore cosf = 0, while at 0° and 180° y = 0 and therefore sinf = 0. We then get 
Angle  sinf = y/r  cosf = x/r 

0°  0  +1 
90°  +1  0 
180°  0  1 
270°  1  0 
360°  0  +1 
Of course, f = 0° and f = 360° represent the same position of r, namely, along the positive branch of the xaxis. Below is the actual plot of cosf:

Next Stop (optional): #M11 Deriving sin(a+b), cos(a+b)
Author and curator: David P. Stern