Answer: x = sqrt(3)/(2sqrt(7)) is the value satisfying domain and equation.
- A x = 1/(2sqrt(7)), obtained by dropping the sqrt(3) factor in the derivation
- B x = sqrt(3)/(2sqrt(7)) is the value satisfying domain and equation
- C x = 1/2, which fails the domain constraint required for sin-1(2x)
- D x = 1/sqrt(7), obtained from an algebra slip in clearing the radical
Correct answer: B. x = sqrt(3)/(2sqrt(7)) is the value satisfying domain and equation
Explanation: Setting sin-1(2x) = pi/3 - sin-1(x) and taking sine of both sides with the addition formula leads, after squaring and simplifying (7x<sup>2</sup> = 3/4 form), to x = sqrt(3)/(2sqrt(7)), which satisfies both the equation and domain |2x|<=1.
The graph of sin⁻¹x is confined to a narrow domain [-1,1] (since sine itself only takes values in that range) and range [-π/2,π/2] (the principal value branch chosen to make sine one-one and invertible there).
Concept context
Restricting trig functions to make them invertible, the principal value branches of sin-inverse, cos-inverse, tan-inverse, and friends, and the key identities relating them.