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The number of solutions of the equation $\sqrt[3]{{\sin \theta - 1}} + \sqrt[3]{{\sin \theta }} + \sqrt[3]{{\sin \theta + 1}} = 0$ in $[0,4\pi]$ is
A
$2$
B
$4$
C
$5$
D
$6$
Solution
$(\sin \theta-1)^{1 / 3}+(\sin \theta+1)^{1 / 3}=-(\sin \theta)^{1 / 3}$
$2 \sin \theta+3\left(-\cos ^{2} \theta\right)^{1 / 3}(-\sin \theta)^{1 / 3}=-\sin \theta$
$3\left(\cos ^{2} \theta\right)^{1 / 3} \cdot(\sin \theta)^{1 / 3}=-\sin \theta$
$(\sin \theta)^{1 / 3}\left(3\left(\cos ^{2} \theta\right)^{1 / 3}+(\sin \theta)^{2 / 3}\right)=0$
$\sin \theta=0 \Rightarrow 5 \text { roots for } \theta \in[0,4 \pi]$
Standard 11
Mathematics
