If 0

Question

(A) We use the Arithmetic Mean-Geometric Mean $(AM \ge GM)$ inequality for four positive terms: $\sin 	heta, \sin 	heta, \sin 	heta$,and $\csc^3

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) We use the Arithmetic Mean-Geometric Mean $(AM \ge GM)$ inequality for four positive terms: $\sin 	heta, \sin 	heta, \sin 	heta$,and $\csc^3 	heta$.$\frac{\sin 	heta + \sin 	heta + \sin 	heta + \csc^3 	heta}{4} \ge \sqrt[4]{(\sin 	heta \cdot \sin 	heta \cdot \sin 	heta \cdot \csc^3 	heta)}$$\frac{3\sin 	heta + \csc^3 	heta}{4} \ge \sqrt[4]{(\sin^3 	heta \cdot \frac{1}{\sin^3 	heta})}$$\frac{3\sin 	heta + \csc^3 	heta}{4} \ge \sqrt[4]{1}$$3\sin 	heta + \csc^3 	heta \ge 4 	imes 1$Thus,the minimum value is $4$.

If $0 < \theta < \pi$,then the minimum value of $3\sin \theta + \csc^3 \theta$ is

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