(M-10) Trigonometry beyond 90°

"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 ("x-axis"). In cartesian coordinates a second "y-axis" 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 x-axis and the "radius" r = OA, measured counter-clockwise.

Since the figure OAPB is a rectangle, the distance AP also equals y. Therefore

sinf = y/ r
cosf = x/ 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
y = r sinf

These relations allow (x,y) to be calculated when (r,f) are given. To go in the opposite direction--given (x,y), find (r,f)--one notes that in the triangle OAP, by Pythagoras

x² + y² = r²

Therefore, given (x,y), r can be calculated, and then (sinf, cosf) can be derived as before by

sinf = y/ r
cosf = x/ 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 right-angled 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 re-define sinf and cosf for such larger angles! The equations are

sinf = y/ r
cosf = x/ 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 right-angled 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:

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 direction--clockwise--can 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

sin²f + cos²f = 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

Of course, f = 0° and f = 360° represent the same position of r, namely, along the positive branch of the x-axis. Below is the actual plot of cosf: