Ordinarily the sine of x radians and the sine of x degrees are very different numbers. Having your calculator in radian mode when it should be in degree mode, or vice versa, results in a major error.
But sometimes it doesn’t matter. A trivial example is when x = 0. A more interesting example is
x = 180π/(180 + π) = 3.08770208….
For that value of x,
sin(x) = sin(x°).
In this article I’ll use the common convention of using radians by default and denoting degrees with ° as above.
Note that
x° = πx/180
and so we are interested in solutions to the equation
sin(x) = sin(πx/180)
Now two angles A and B have the same sine if they differ by a multiple of 2π, or if they’re supplementary (i.e. A = π − B), or both. To put it another way, if A and B have the same sine, they are either equal mod 2π or supplementary mod 2π. This means that
sin(x) = sin(πx/180)
if and only if
x = πx/180 + 2πk
or
x = π − πx/180 + 2πk
for some integer k.
Therefore all solutions have the form
x = 360πk/(180 − π)
or
x = 180π(2k + 1)/(180 + π).
Alternative solution
The derivation above is correct, but it occurred to me later that a simpler argument would be to use the identity
sin(A) − sin(B) = 2 cos((A + B)/2) sin((A − B)/2).
Thus A and B have the same sine if
cos((A + B)/2) = 0
or if
sin((A − B)/2) = 0.
These two possibilities correspond to the two families of solutions above.
Density
When reduced modulo 2π, both families are dense in [0, 2π]. This means that for every y in [−1, 1], there is a number x such that
sin(x) = sin(x°) ≈ y
and we can make the approximation as good as we’d like.
Example 1
For example, today is July 22, so let’s set y = 0.722. We’d like to find a value of x such that the sine of x radians and the sine of x degrees both approximately equal 0.722. And let’s say our approximation tolerance is ε = 0.0001.
We can search for a value of x in the first family of solutions by looking for a value of k with
| sin(360πk/(180 − π)) − 0.722 | < 0.0001
and the smallest such k is 96343 and so
x = 360×96343 π/(180 − π) = 616093.78713621…
will do, and sin(x) = 0.72191…
Example 2
Now let’s set y = 0.2026 and look for a solution in the other family of solutions, and this time let’s set ε = 10−6. The smallest value of k such that
| sin(180π(2k + 1)/(180 + π)) − 0.2026 | < 10−6
is k = 741141. Then
sin( 4576848.310950611 ) = sin( 4576848.310950611° ) = 0.202600139…