Question

if cosec theta -cot theta = 1/3 then theta lies in which quardrant​

Answer 1

Answer:

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Answer 2

Answer:

θ ∈ Q1

Step-by-step explanation:

$Given\\\\cosec \theta -cot \theta = 1/3\\\\FORMULAE\\\\cosec\theta=\frac{1}{sin\theta}\\\\tan\theta=\frac{1}{cot\theta}\\\\cot\theta=\frac{cos\theta}{sin\theta}\\\\sin2\theta=2sin\theta cos\theta\\\\cos2\theta=1-2sin^2\theta\\\\Now\\\\\\cosec \theta -cot \theta = 1/3\\\\\frac{1-cos\theta}{sin\theta}=\frac{1}{3}\\\\\frac{2sin^2\frac{\theta}{2}}{2sin\frac{\theta}{2}cos\frac{\theta}{2}}=\frac{1}{3}\\\\tan\frac{\theta}{2}=\frac{1}{3}\\\\tan\frac{\theta}{2}>0\\\\=>0<\frac{\theta}{2}<\frac{\pi}{2}$

$and\\\\\pi<\frac{\theta}{2}<\frac{3}{2}\pi\\\\Now\\\\=>0<\theta<\pi\\\\=>2\pi<\theta<3\pi\\\\That \;\;0<\theta<\pi\\\\We\;got\\tan\frac{\theta}{2}=\frac{1}{3}\\\\=>sin\frac{\theta}{2}=\frac{1}{\sqrt{10}}\\\\=>cos\frac{\theta}{2}=\frac{3}{\sqrt{10}}\\\\=>sin\theta=\frac{6}{10}=\frac{3}{5}\\\\=>cos\theta=\frac{4}{5}\\\\=>cosec\theta=\frac{5}{3}\\\\=>cot\theta=\frac{4}{3}\\\\Since\;\;cot\theta>0\\\\0<\theta<\frac{\pi}{2}\\\\=>\theta \;\;belongs\;\;to\;\;Q_1$

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