Instead of thinking of them as ratios to be mindlessly memorized, let's return to what they represent on the unit circle, and on the unit circle only, since that's where they come from. Let's drop the hypothenuse. After all, it's just a fancy name for RADIUS. To be more precise, we could call it the UNIT RADIUS, but most of the time we simply can refer to it as the RADIUS, since we know it's equal to 1. If we push a wheel a full TURN (instead of 2 PI, or TAU) , it will advance always the same distance. Let's call that distance the TRAVEL (instead of circumference). Let's call ANGLE a fraction of a TURN (instead of RADIANS), and ARC the portion of the unit circle that has made contact with the ground. And so on... The sinus on the unit circle represents the RISE of the ANGLE. The cosinus represents the RUN of the ANGLE. The tangent is now called the THROW of the ANGLE. The cotangent is the BACKTHROW or BTHROW of the ANGLE The secant is the horizontal projection or HPROJ of the ANGLE the cosecant is the vertical projection or VPROJ of the ANGLE sin(90 degrees) = sin(PI/2) = RISE(1/4 TURN) = RISE(0.25) = 1 cos(45 degrees) = cos(PI/4) = RUN(1/8 TURN) = RUN(0.125) = SQRT(2) tan(45 degrees) = tan(PI/4) = THROW(1/8 TURN) = THROW(0.125) = 1 The ARC on the unit circle is equal to the ANGLE in fraction of a TURN. The CHORD on the unit circle is equal to TWO times the RISE of HALF the ANGLE. The sum of the ANGLES of a TRIANGLE equals HALF A TURN If you want to compute results on a circle that bigger or smaller than the unit circle, you need to multiply all the distances by the proportion of its radius compared to the unit circle RADIUS.
