I didn't follow every last instruction, but it looks like it used a 91 (or maybe 92)-element LUT of 4-byte floats, split into 2 halves. This allows direct look-up of the SIN of every angle in the first quadrant, that is 0 to 90 degrees, plus the value for the next 1-degree angle.
It looks like after looking up the two precalculated values, linear interpolation is then performed between them.
(Also, the FP numbers are in a format that must have been easy to do software arithmetic on. Leading byte is exponent, with bias of 126; next 3 bytes are sign and mantissa, with no implied 1-bit. This has just a hair less precision than a modern 32-bit 'float'. 7A 47 7C 2D is the entry for SIN(1), and is equal to 0.017452407... while double precision SIN(1) is 0.0174524064372835...)
Next, I bodged together a version of what I assume the linear interpolation step does. In both double precision and simulated LOGO precision, the maximum relative error is about .00005 and the maximum absolute error is about .000045, which would be awful hard to detect when plotting to an entire display with about as many pixels as an application icon on your modern high-dpi telephone. On the other hand, that's only 4 correct digits.
This is an interesting contrast to the Commodore BASIC implementation of sin/cos that I'm familiar with. It used a fairly high degree polynomial, which might have gotten a few more accurate digits (5 digits for sin(1.5 degrees)!) but is sure to be quite a bit slower than 2 table lookups, a multiply, and a few add/subtracts. (plus range reduction)
0.0261759515 Estimated Apple LOGO SIN(1.5 degrees)
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0.026176948307873153 Modern Double Precision SIN(1.5 degrees)
0.0261769483 Commodore 64 BASIC SIN(1.5 degrees)
^