Possible to analytically find the sine of any angle?

Is it possible to find an exact answer for something like sine(46 degrees)? I understand how to use the sum formula for sine(a+b) to get something like sine(75). I also see how you could use the sine(a/2) formula to get something like sine (22.5) and keep halving it to get lower and lower.

But if one only knows exact values for the sine of 30,45, and 60, the sum formula and the halving formula, wouldn't it be impossible to find an exact answer for the sine(23)?