When sine of x degrees equals sine of x radians

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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 | 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 | −6

is k = 741141. Then

sin( 4576848.310950611 ) = sin( 4576848.310950611° ) = 0.202600139…

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John D. Cook, PhD

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