(5cotθ+cosecθ)/(5cotθ-cosecθ)=7/3 হলে, cosθ এর মান কত

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প্রশ্ন

\dfrac{5\cot\theta+cosec\theta}{5\cot\theta-cosec\theta}=\dfrac{7}{3}

হলে, \cos\theta এর মান কত?

উত্তর

\cos\theta = \dfrac{1}{2}

ব্যাখ্যা / সমাধান

\dfrac{5\cot\theta+cosec\theta}{5\cot\theta-cosec\theta}=\dfrac{7}{3}

\Rightarrow \dfrac{5\left(\dfrac{\cos\theta}{\sin\theta} \right)+\dfrac{1}{\sin\theta}}{5\left( \dfrac{\cos\theta}{\sin\theta} \right)-\dfrac{1}{\sin\theta}}=\dfrac{7}{3}

\Rightarrow \dfrac{\dfrac{5\cos\theta + 1}{\sin\theta}}{\dfrac{5\cos\theta - 1}{\sin\theta}}=\dfrac{7}{3}

\Rightarrow \dfrac{5\cos\theta + 1}{\sin\theta} \times {\dfrac{\sin\theta}{5\cos\theta - 1}}=\dfrac{7}{3}

\Rightarrow \dfrac{5\cos\theta + 1}{5\cos\theta - 1}=\dfrac{7}{3}

\Rightarrow 3(5\cos\theta + 1)=7(5\cos\theta - 1)

\Rightarrow 15\cos\theta + 3=35\cos\theta - 7

\Rightarrow 20\cos\theta = 10

\Rightarrow \cos\theta = \dfrac{1}{2}

[\cos\theta = \cos45^\circ]

\therefore \cos\theta এর মান \dfrac{1}{2}

প্রয়োজনীয় সূত্রাবলী:

\cot\theta = \dfrac{\cos\theta}{\sin\theta}

cosec\theta=\dfrac{1}{\sin\theta}

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