Quadratic EquationHard
Question
If $sin10^{\circ}$ is a root of the equation $4{ax}^{3} - 3ax + b = 0$, where $a,b$ are real parameters, $a \neq 0$, then the remaining two roots are
Options
A.$sin130^{\circ}$
B.$sin40^{\circ}$
C.$sin250^{\circ}$
D.$sin200^{\circ}$
Solution
$\frac{b}{a} = 3x - 4x^{3}$
$$\begin{matrix} \text{~Put~}x = sin\theta\ \Rightarrow \ sin3\theta & & = \frac{b}{a} \\ sin10^{\circ}\text{~is a root~} \Rightarrow & & sin30^{\circ} = \frac{b}{a} = \frac{1}{2} \\ \therefore & & sin3\theta = sin\left( 360^{\circ} + 30^{\circ} \right) = sin\left( 720 + 30^{\circ} \right) \\ \Rightarrow & & 3\theta = 390^{\circ},750^{\circ} \\ \Rightarrow \ & & \theta = 130^{\circ},250^{\circ} \end{matrix}$$
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