If 11 \sin \theta+60 \cos \theta=61; 0 \lt \theta \lt 90^{\circ}, then what is the value of \sqrt{660(\tan \theta+\cot \theta)}?

Explanation

Dividing by 61 gives \frac{11}{61}\sin\theta + \frac{60}{61}\cos\theta = 1. Comparing with \sin^2\theta + \cos^2\theta = 1, we get \sin\theta = \frac{11}{61} and \cos\theta = \frac{60}{61}. Then \tan\theta + \cot\theta = \frac{11}{60} + \frac{60}{11} = \frac{3721}{660}. The required value is \sqrt{660 \times \frac{3721}{660}} = \sqrt{3721} = 61.

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