APL (Dyalog Unicode), 56 55 bytes

d,↑0 60 60∘⊤¨¨⌊.5+c,¨2×0,⍨¯2-/c←432e3×1○360÷⍨○d←2÷⍨⍳360

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d←2÷⍨⍳360 all the angles \$\theta\$ from 0.5 to 180.
1○ Sine of 360÷⍨○ \$\theta \over 2\$ in radians.
432e3× Multiply by \$432000 = 60×60×120\$. This gives the chord length adjusted such that both sexagesimal digits are in the integer part.

¯2-/ Differences of adjacent values, for the "sixtieths" column.
0,⍨ Append a 0, this hardcodes the sixtieths for 180°.
2× Multiply each value by \$2 = 60 ÷ 30\$.

c,¨ Pair up each chord length with corresponding sixtieths value. ⌊.5+ Round all value to the nearest integer. 0 60 60∘⊤¨¨ Convert each value from "base 60", where the first of three digit is unbounded.

↑ Remove one layer of nesting to get a table.
d, prepend the angles on the left of the table.

APL (Dyalog Unicode), 56 55 54 bytes

A full program printing the table

d,0 60 60∘⊤¨⌊.5+↑c,¨2×0,⍨¯2-/c←432e3×1○360÷⍨○d←2÷⍨⍳360

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d←2÷⍨⍳360 all the angles \$\theta\$ from 0.5 to 180.
1○ Sine of 360÷⍨○ \$\theta \over 2\$ in radians.
432e3× Multiply by \$432000 = 60×60×120\$. This gives the chord length adjusted such that both sexagesimal digits are in the integer part.

¯2-/ Differences of adjacent values, for the "sixtieths" column.
0,⍨ Append a 0, this hardcodes the sixtieths for 180°.
2× Multiply each value by \$2 = 60 ÷ 30\$.

↑c,¨ Pair up each chord length with corresponding sixtieths value. This results in a matrix with two columns. 
⌊.5+ Round all values to the nearest integer. 
0 60 60∘⊤¨ Convert each value from "base 60", where the first of three digit is unbounded.

d, prepend the angles on the left of the table.

APL (Dyalog Unicode), 56 bytes

↑d,¨0 60 60∘⊤¨¨⌊.5+c,¨2×0,⍨¯2-/c←432e3×1○360÷⍨○d←2÷⍨⍳360

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APL (Dyalog Unicode), 56 55 bytes

d,↑0 60 60∘⊤¨¨⌊.5+c,¨2×0,⍨¯2-/c←432e3×1○360÷⍨○d←2÷⍨⍳360

Try it online!

d←2÷⍨⍳360 all the angles \$\theta\$ from 0.5 to 180.
1○ Sine of 360÷⍨○ \$\theta \over 2\$ in radians.
432e3× Multiply by \$432000 = 60×60×120\$. This gives the chord length adjusted such that both sexagesimal digits are in the integer part.

¯2-/ Differences of adjacent values, for the "sixtieths" column.
0,⍨ Append a 0, this hardcodes the sixtieths for 180°.
2× Multiply each value by \$2 = 60 ÷ 30\$.

c,¨ Pair up each chord length with corresponding sixtieths value. ⌊.5+ Round all value to the nearest integer. 0 60 60∘⊤¨¨ Convert each value from "base 60", where the first of three digit is unbounded.

↑ Remove one layer of nesting to get a table.
d, prepend the angles on the left of the table.