The volume contains values of trigonometric functions for every thousandth of a degree, to seven places. The first table give sines and cosines, the second tangents and cotangents. Information in the ...

Both the sine and cosine functions have outputs between -1 and 1 (because the x and y values can be negative). You can think of these trig functions as a sort of "lookup table." ...

So what? Divide this by 2 and you get 0.80901699437494..., which agrees with our calculation of the sine of 666°, up to a sign. That's not a proof of course, since we only have 14 decimal places ...

Part of that is the way it's taught. Students are taught the "unit circle" and its relationship to trigonometry, but many fail to make the leap on how crucial circles are for trig functions.