(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,φ).

   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

sin φ = y/r
cos φ = 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 cos φ
y = r sin φ

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

x² + y² = r²

   Therefore, given (x,y), r can be calculated, and then (sin φ, cos φ) can be derived as before by

sin φ = y/r
cos φ = 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 φ).

   However, there remains a problem. The angle φ as defined above can go from 0 to 360°, but (sin φ, cos φ) 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 φ is larger than 90 degrees, and such angles never appears in any right-angled triangle. What sort of meaning can (sin φ, cos φ) have for φ larger than 90 degrees?

   There is a simple solution, though: use the above equations to re-define sin φ and cos φ for such larger angles! The equations are

sin φ = y/r
cos φ = x/r

   They are now viewed as new definitions of the sine and cosine, for the polar angle φ given by x and y (a slightly different way of formulating this definition is described further below). 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 φ 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 φ. Together, these extensions define (sin φ, cos φ) for any angle φ, positive or negative, of any size.

The relation derived from Pythagoras' theorem

sin²φ + cos²φ = 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 cos φ = 0, while at 0° and 180° y = 0 and therefore sin φ = 0. We then get

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

A Slightly Different Definition: the Unit Circle

   Many trigonometry texts define the sine and cosine slightly differently, using the so-called unit circle. That is a circle whose radius is chosen to equal 1 unit (in whatever units we measure). We draw the circle so that its center is at the origin O of an (x,y) set of coordinates, and imagine a movable radius OB making an angle a with the x-axis.

   Then the distance AB ("the sine line") equals sin a, and the distance OA ("the cosine line") equals cos a. The line CD, cut off the tangent to the circle by the extension of OB, is the "tangent line," it equals tan a and explains why the name "tangent" was given to this quantity.

    Suppose the radius rotates to position OB', so that its angle with the x-axis is b, larger than 90°. Then the sine line A'B' still has positive length, since it is above the horizontal axis. However the cosine line OA' is to the left of the origin, so its length--which gives the cosine--must be counted as negative. The slice cut off the tangent is now the line CD' produced by the extension OD' of the rotating radius, and its length is counted as negative, too.

By letting the radius rotate all around the circle and measuring distance--with the understanding that anything extending below the horizontal axis or to the left of the vertical one is negative--we find that the sine line, the cosine line and the tangent line always give the correct functions. It is really not different from the previous definition: but if you ever wondered, how the term "tangent" entered trigonometry, now you know.