#2 sin x = cos(x/3) #
This is a pretty tough one.
Let's start by setting #y=x/3# so #x=3y# and substituting. Then we can use the triple angle formula:
#2 sin(3y) = cos y#
#2 (3 sin y - 4 sin ^3 y ) = cos y#
Let's square so we write everything in terms of #sin^2 y#. This will likely introduce extraneous roots.
# 4 sin ^2y (3 - 4 sin ^2y)^2 = cos ^2 y = 1 - sin^2 y #
Let #s= sin ^2 y#. Squared sines are called spreads in Rational Trigonometry.
#4 s(3 - 4s)^2 = 1 - s#
#4 s( 9 - 24 s + 16 s^2) = 1 - s #
# 64 s^3 - 96 s^2 + 37 s - 1 = 0 #
That's a cubic equation with three real roots, candidates for the squared sines of #3x.# We could employ the cubic formula, but that will just lead to some cube roots of complex numbers that aren't particularly helpful. Let's just take a numerical solution :
#s≈0.66035 or s≈0.029196 or s≈0.81045#
#x = 3y = 3 arcsin (pm sqrt{s}) #
Let's work in degrees. Our potential approximate solutions are:
# x = 3 arcsin(\pm sqrt{ 0.66035 } ) approx \pm 163.058^circ or \pm 703.058^circ#
# x = 3 arcsin(\pm sqrt{ 0.029196 } ) approx \pm 29.5149^circ or \pm 569.51^circ#
# x = 3 arcsin(\pm sqrt{ 0.81045 } ) approx \pm 192.573^circ or pm 732.573^circ #
Let's see if any of those work. Let #e(x)=2 sin x - cos ( x/3)#
#e( 163.058^circ) approx 0.00001 quad # that's a solution.
#e(-163.058^circ) approx -1.17 quad # not a solution.
Clearly at most one of a #\pm# pair will work.
Ten more to go.
#e( 703.058^circ ) approx 0.00001 quad sqrt #
#e( -703.058 ^circ) quad # nope
#e( 29.5149^circ ) approx 10^{-6} quad sqrt#
#e( -29.5149^circ ) quad # nope
#e( 569.51^circ ) approx 10^{-4} quad sqrt#
#e( -569.51^circ ) quad # nope
#e( 192.573^circ ) approx -.87 quad # nope
#e( -192.573^circ ) approx 0.00001 quad sqrt #
#e( 732.573^circ ) approx -.87 quad # nope
#e( -732.573^circ ) approx 0.00001 quad sqrt #
The arcsin comes with a #+ 360^circ k#, and the factor of three makes it #1080^circ k.#
OK, our approximate solutions are:
# x = { 163.058^circ, 703.058^circ, 29.5149^circ, 569.51^circ , -192.573^circ , -732.573^circ } + 1080^circ k quad # for integer #k#.
