In the file, of which this is a continuation, you constructed a UNIT CIRCLE with CENTER AT ORIGIN OF A CARTESION COORDINATES SYSTEM; you drew a RADIAL IN THE FIRST QUARANT and labeled it point on the CIRCLE as (x,y) or (cos q, sin q), where q denotes the ANGLE the radial makes to the X-axis. Let's proceed from this.

1. Put radials in the 2nd, 3rd, and 4th quadrants, labeling in a similar manner, along with the projections onto the X-axis, creating right triangles. Of course, the second usage of q denotes an angle greater than 90° and less than 180°; the third usage denotes an angle greater than 180° and less than 270°; the fourth usage denotes an angle greater than 270° and less than 360°.
2. In QUADRANT I, as the radial goes from 90°, the projected positive-y segment increases in length, so the sine INCREASES with q increase in this quadrant; and the projected positive-x segment decreases in length, so the >cosine DECREASES with q increase in this quadrant.
3. In QUAD. II, as the radial goes from 180°, the projected positive-y segment decreases in length, so the sine DECREASES with q increase in this quadrant; and the negative-x segment increases in length, so the cosine INCREASES NEGATIVELY in this quadrant.
4. In QUAD. III, as the radial goes from 180° to 270°, the projected negative-y segment increases in length, so the sine INCREASES NEGATIVELY in this quadrant; and the negative=x segment decreases in length, so the cosine DECREASES NEGATIVELY in this quadrant.
5. In QUAD. IV, as the radial goes from 270° to 360°, the projected negative-y segment decreases in length, so the sine DECREASES NEGATIVELY TOWARD ZERO in this quadrant; and the positive-x segment increases in length, so the cosine INCREASES TOWARD 1 in this quadrant.

Dig?

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GRAPHING TANGENT AND SECANT FUNCTIONS

The triangles-in-circle graph the change of the sine and cosine functions in the four quadrants. We could derive the change of the tangent function from the changes of the sine and cosine functions, using the definition of tangent in terms of sine and cosine. But this is tedious. Fortunately, we can similarly graph the tangent function. (This is something I learned a few years after learning the Cauchy device.)

1. Given the triangles in each quadrant of the circle.
2. Draw a TANGENT on the right of the circle, perpendicular to the X-axis; and, similarly, a TANGENT on the circle's left.
3. Extend the radial lines, in each quadrant, to intersect these tangent lines, creating four larger right triangles, which include the the ones first constructed. The lengths of the OPPOSITE sides of these larger triangles represents the tangent function in each quadrant.
4. 1st Quadrant: As the extended radial sweeps (like a clock-hand) from 0° to 90°, the tangent segment (triangle's OPPOSITE side) increases positively from 0 to infinity, measuring the values of the TANGENT function in Quadrant I. 2nd Quadrant: as extended radial sweeps from 90° to 180°, the tangent segment DECREASES through positive values fom infinity to zero, measuring the values of the TANGENT function in Quadrant II. 3rd Quadrant: as the extended radial sweeps from 180° to 270°, the tangent segment extends NEGATIVELY from 0 to NEGATIVE INFINITY, measuring the values of the TANGENT function in Quadrant III. 4th Quadrant: As the extended radial sweeps from 270° to 360°, the tangent segment INCREASES through NEGATIVE values form NEGATIVE INFINITY to ZERO, measuring the values of the TANGENT function in QUADRANT IV.

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   These extended triangles provide a measure of another trigonometric function.
   1. Where the original set of triangles has a CONSTANT HYPOTENUSE of ONE UNIT LENGTH, each extended triangle has a CONSTANT ADJACENT side of ONE UNIT LENGTH. And we know that the TANGENT function measures the ADJACENT SIDE. What about each HYPOTENUSE of these extended triangles?
   2. Apply the PYTHAGOREAN THEOREM to find: tan² q + 1² = (hypotentuse length)-squared. Does this remind you of an idenity we studied? Yes: tan² + 1 = sec². So each HYPOTENUSE of the extended triangle in each measures, by its changes, the SECANT function in each of these quadrants.
   Challenge: Work the rest of it out.

   Challenge: The COSECANT function gets measured in this, somehow. I knew once. But lost the work. See if you can explicate it.

   The late Bertrand Russell (1872-1970) said that axiomatics "has all the advantages" over generatics "that theft has over honest labor" -- assuming what should be generated from a basis of elements. Don't you feel better for working out so much of TRIGONOMETRY?