If 8\sin \theta - \cos \theta = 4, where 0 < \theta < \pi/2, then what is \operatorname{cosec} \theta equal to?

Explanation

Given 8\sin \theta - 4 = \cos \theta. Squaring both sides: 64\sin^2 \theta - 64\sin \theta + 16 = 1 - \sin^2 \theta. This simplifies to 65\sin^2 \theta - 64\sin \theta + 15 = 0. Factoring yields (5\sin \theta - 3)(13\sin \theta - 5) = 0. Since \theta is such that options match \operatorname{cosec} \theta, if \sin \theta = 3/5, then \operatorname{cosec} \theta = 5/3.

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