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sin^-1(2x√1-x^2)+sin^-1(3x -4x^3)=-pi/3 the value of x
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Notice that due to the formula sin(2a) = 2*sin(a)*cos(a),
you have arcsin((2x*sqrt(1-x^2)) = 2a, where sin(a) = x.
Similarly, due to the formula sin(3a) = 3*sin(a) - 4*sin^3(a),
you have arcsin(3x-4x^3) = 3a, where sin(a) = x.
Therefore, the given equation
sin^(-1)(2x*sqrt(1-x^2)) + sin^(-1)(3x -4x^3) = -pi/3
is equivalent to
2a + 3a = -pi/3, where sin(a) = x,
or
5a = -pi/3, sin(a) = x,
a = = = -12°.
It implies x = . It is the "exact" formula and the "exact" answer.
To get the numerical value, use your calculator or table of sinus: x = -0.20791169081... = -0.2079 (rounded).
CHECK. 2x*sqrt(1-x^2) = = -0.406715; arcsin(-0.406715) = -0.418855117.
3x -4x^3 = 3*(-0.2079) - 4*(-0.2079)^3 = -0.587756244; arcsin(-0.587756244) = -0.628282675.
-0.418855117 - -0.628282675 = -1.047137791.
From the other side, = = -1.04719755.
Both values, -1.047137791 and -1.04719755, are close enough, confirming validity of the answer.
Solved.