DEVELOP GENERATIVE TRIGONOMETRY

Why is TRIG so great?

  1. APPLICATION: TRIGONOMETRY helps us measure the world and guide us through it.
  2. THEORY: TRIGONOMETRY provides a REPERTORY of "identities", such as sin2 q + cos2 q = 1 or sin (a + b) = sin a cos b + cos a sin b.
Some mathematician (I fergit who) said that 95% of the uses of TRIGONOMETRIC IDENTITIES has nothing to do with triangles or other applications. He didn't refer to the ALGORITHM of "BYPASS", but that's how they are used. TO BYPASS A DIFFICULT PROBLEM BY USING A TRIG-IDENTITY TO TRANSFORM TO A SOLVEABLE PROBLEM, AND USE A TRIG-IDENTITY TO TRANSFORM THE ANSWER BACK TO THE NEEDED FORM. This is the main use in the INTEGRAL CALCULUS. Given an "integrand" as a DERIVATIVE-FORM, but you don't know or immediately recognize its ANTIDERIVATIVE to solve the given problem, you use an TRIG-IDENTITY to transform to one you can handle.

I'm telling you this because you should know it and be guided by it in the future. And also because the GENERATIVE method I describe CONNECTS with this ALGORITHM.

HOW I LEARNED NOT TO FEAR TRIGONOMETRY

My last precollege math course was 9th Grade (Numerical) Algebra. As explained elsewhere, because I was "working class" I was denied math and science and languages classes in High School in Springfield, MO, in the 30's. Later, in Tulsa, OK, finishing high school, I was allowed to take physics and chemistry, but no math.

This created a problem for many of my generations who went into the Service during World War II, and needed math, science, languages. As a Army Air Corps weather observer, my going to Weather Forecasting School was delayed by my scores on qualifying tests were diminished by trigonometry errors. I finally made it, but developed a fear of the subject.

As noted elsewhere, when I enter Columbia University in 1948, under the "G. I. Bill", I wanted to major in English, then Journalism, and become a science writer. But the Veterans Administration would'n't allow it, and forced my to major in physics. I should not have been registered at Columbia under this major, since I didn't have the prerequisites. But Columbia wanted the Government money. My adviser put me into College Algebr and Calculus I, simply because they were open. This began 8 years of fear and loathing in university education! I survived, with traumas.

In 1955, I went with my wife and 1-year old son, Tim, to teach math and physics in Instituto Polytechnico (later Inter American University of Puerto Rico), San Germán, PR. Guess what? The first two courses I had to teach were Trigonometry and Physics Laboratory, the subjects I hated most!

Happily, I soon discovered, in a physics magazine, the GENERATIVE INTRODUCTION TO TRIGONOMETRY developed by the great 19th century mathematician, Augustin Cauchy (1789-1857). It vaniquished my fear! Hallelujah!

GENERATING THE TRIGONOMETRIC FUNCTIONS
  1. In a plane Cartesian coordinate system, plot a UNIT CIRCLE WITH CENTER AT THE ORIGIN. This provides the "four quadrants" as REFERENCE POINTS IN TRIG. (1st quadrant: positive x, y coordinates; 2nd: negative x, positive y coordinates; 3rd: negative x,y coordinates; 4th: positive x, negative y coordinates.)
  2. Draw a radial from the center to some point of the 1st quadrant arc above the X-axis. As standard, label this point, (x,y).
  3. This creates an angle between the X-axis and this radial; label this angle q.
  4. Now modify the point label, (x,y), writing it as (x = cos q, y = sin q), or simply (cos q, sin q).
  5. Project this point onto the X-axis, creating a RIGHT TRIANGLE, whose OPPOSITE SIDE is this PROJECTION; whose ADJACENT SIDE is the segment along the X-axis from center to projection onto X-axis; the HYPOTENUSE is the radial line (of length 1 unit!).
  6. EVERYTHING GENERATES FROM THIS!
You recall the standard definition of the TRIG FUNCTIONS in terms of a RIGHT TRIANGLE:
  1. cos q = ADJACENT/HYPOTENUSE. Here, cos q = x/r = x/1 = x. That is, THE TRIANGLE'S ADJACENT SIDE (segment from center to base of projection) equals cos q.
  2. sin q = OPPOSITE/HYPOTENUSE. Here, sin q = y/r = y/1 = y. That is, THE TRIANGLE'S OPPOSITE SIDE (projection segment) equals sin q.
  3. Given the sine and cosine functions, you can GENERATE the other four standard ones ("nonstanard" ones such as "haversine" have navigational uses) from sine and cosine.
    • tan q = sin q/cos q;
    • cot q = 1/tan q = cos q/sin q.
    • sec q = 1/cos q;
    • csc q = 1/sin q.
  4. The RIGHT TRIANGLE then gives you the most important of the TRIGONOMETRIC IDENTITES, namely, sin2 q + cos2 q = 1. Do you see it? Apply the PYTHAGOREAN THEOREM to this RIGHT TRIANGLE in your construction: (opposite side length)-squared = x2 = sin2 q; (adjacent side length)-squared = y2 = cos2 q; (hypotenuse length)-squared = (radius)-squared = 12 = l. Substitute, and you have th desired identity.
  5. All other identities can be DERIVED from this and the definitions of the trigonometric functions. Thus,
    • given sin2 q + cos2 q = 1;
    • divide this through by cos2 q;
    • apply to this, tan q = sin q/cos q; and tan2 q + 1 = sec2.
Challenge, can you, similarly, derive the identity: cot2 q + 1 = csc2?

You can derive all other TRIGONOMETRIC IDENTITIES in a similar fashion. Admittedly, it's often tedious, at best, and confusing much of the time, such as trying to derive sin (a + b) = sin a cos b + cos a sin b I did it some 40 years ago, but lost the work.

I'm merely asking you to do enough of them to convince yourself that ALL THE TRIGONOMETRIC IDENTITIES ARE AS CONNECTED AS CHARLOTTE'S WEB!

Remember the equations for CIRCUMFERENCE OF A CIRCLE?

		C = 2pr.
The CIRCUMFERENCE is THE FULL ARC of the CIRCLE. Let's DERIVE from this an EQUATION of any sort of ARC. And then obtain another useful measure:
  1. REPLACE C by arc:
    
		arc = 2pr.
  2. And the 2p on the right side of both equations is an ANGLE, which we can symbolize by q:
    
		arc = qr.
  3. This is a useful general formula. Butone more step yields a useful measure. We've been using the UNIT CIRCLE: RADIUS R = 1. REPLACING the radius in the above formula, we have:
    
		arc = q.
  4. This says that ANY ARC OF THE UNIT CIRCLE IS MEASURED BY (a.k.a. EQUIVALENT TO) THE ANGLE IT SUBTENDS. But this is a MEASURE, which has a special name, "radian". Since the full ARC of the CIRCLE equals 2p in ANGLE (equaling ARC ON UNIT CIRCLE), we can obtain THE UNIT RADIAN by DIVIDING both sides of the previous equation by 2p:
    
		arc/2p = 2p/2p.
  5. But when that ANGLE is THE FULL CIRCULAR ANGLE, namely, 2p, the right side above becomes 2p/2p = 1. Then, finally, we have our MEASURE DEFINITION:
    
		arc/2p = 1 radian.
  6. And this means that, to A 7-digit APPROXIMATION, 1 radian » 57.29578°.
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QUANDRANTAL VALUES OF TRIG FUNCTIONS.