When sine of x degrees equals sine of x radians

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 = 180/(180 + ) = 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= x/180

and so we are interested in solutions to the equation

sin(x) = sin(x/180)

Now two anglesA andB have the same sine if they differ by a multiple of 2, or if they're supplementary (i.e. A = - B), or both. To put it another way, ifA andB have the same sine, they are either equal mod 2 or supplementary mod 2. This means that

sin(x) = sin(x/180)

if and only if

x = x/180 + 2k

or

x = - x/180 + 2k

for some integerk.

Therefore all solutions have the form

x = 360k/(180 - )

or

x = 180(2k + 1)/(180 + ).

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 andB 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 2, both families are dense in [0, 2]. This means that for everyy in [-1, 1], there is a numberx 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 sety = 0.722. We'd like to find a value ofx such that the sine ofx radians and the sine of x degrees both approximately equal 0.722. And let's say our approximation tolerance is = 0.0001.

We can search for a value ofx in the first family of solutions by looking for a value ofk with

| sin(360k/(180 - )) - 0.722 | < 0.0001

and the smallest suchk is 96343 and so

x = 360*96343 /(180 - ) = 616093.78713621...

will do, and sin(x) = 0.72191...

Example 2

Now let's sety = 0.2026 and look for a solution in the other family of solutions, and this time let's set = 10^-6. The smallest value ofk such that

| sin(180(2k + 1)/(180 + )) - 0.2026 | < 10^-6

isk = 741141. Then

sin( 4576848.310950611 ) = sin( 4576848.310950611 ) = 0.202600139...

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