F256 SuperBASIC Trigonometry (Foenix F256K2)

trig.bas

This F256 SuperBASIC program calculates sine, cosine, and tangent for any input angle in degrees on the Foenix F256K2.

It uses:

• Quadrant reduction to fold any angle into the first quadrant (0°–90°),
• A truncated Taylor series for sin(x) (up to the x⁷ term),
• Complementary-angle identities to derive cos(x),
• A simple ratio to compute tan(x).

How it works:

1. Input

   • The program asks for an angle in degrees.

2. Quadrant reduction

   • Angles are reduced by subtracting 90° repeatedly until they fall within 0–90°.

3. Series expansion

   • Uses sin(x) ≈ x − x³/3! + x⁵/5! − x⁷/7! with x in radians.

4. Cosine calculation

   • Derived from the complementary angle (cos θ = sin(90°−θ)), with sign adjustments by quadrant.

5. Tangent

   • Computed as sin / cos.

apop4.bas

This F256 SuperBASIC program calculates sine and cosine values for any input angle (with up to two decimal places in degrees) using lookup tables (LUTs) and trigonometric relations.
It is designed for the Foenix F256K2 computer and demonstrates efficient trigonometric calculations without heavy computation.

How it works:

1. Precomputed tables

   • trig#(2,91) contains sine and cosine values from 0° to 90° in steps of 1°.
   • dec#(2,99) contains small incremental values for fractional degrees (0.01° to 0.99°).

2. Quadrant reduction

   • Any input angle d# is mapped into the first quadrant [0°, 90°].
   • The quadrant number is tracked so correct signs can be applied afterwards.

3. Trigonometric relations

   • Integer part of the angle → base sine/cosine values from the main LUT.
   • Fractional part (two decimals) → fine adjustment from the incremental LUT.
   • Formula used (see here):

     sine  = sin_base * cos_frac + cos_base * sin_frac
     cosine = cos_base * cos_frac - sin_base * sin_frac
     

4. Quadrant sign correction

   • Adjusts sin and cos signs according to the quadrant (I–IV).

cordic8.bas

This F256 SuperBASIC program calculates sine, cosine, and tangent for any input angle in degrees (with five decimals) on the Foenix F256K2.

It uses the CORDIC algorithm (Coordinate Rotation Digital Computer), implemented with lookup tables (LUTs) for efficiency.

How it works:

1. Precomputed tables

   • tan#(25) contains tangent values of the CORDIC rotation angles.
   • arctan#(25) contains the corresponding arctangent angles in degrees.
   • cos#(25) contains scaling (cosine) factors used to normalize the final result.

2. Quadrant reduction

   • Any input angle d# is reduced into the first quadrant (0°–90°).
   • A quadrant flag (1–4) is stored so the correct signs can be applied to sine/cosine afterwards.

3. CORDIC iteration

   • Starts with an initial vector (x, y) = (1, 0).
   • Repeatedly rotates this vector by predefined angles using only additions and multiplications by precomputed tangents.
   • Accumulates the scaling factor k# from the cosine LUT.
   • Converges toward the requested angle with 25 iterations.

4. Final normalization

   • After iterations, x and y are scaled by k#.
   • Results:
     • sine# = y
     • cosine# = x
     • tan# = sine# / cosine#
   • Signs are adjusted based on the original quadrant.