Evaluate the principal value of \(\csc^{-1}(-1)\).\(\csc^{-1}(-1)\) का principal value मान निकालिए।
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(-\frac{\pi}{2}\)\(-\frac{\pi}{2}\)
Let \(\theta=\csc^{-1}(-1)\). Then \(\csc\theta=-1\), i.e., \(\frac{1}{\sin\theta}=-1\). So \(\sin\theta=-1\). The principal range for \(\csc^{-1}x\) is \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). In this range, \(\sin\theta=-1\) occurs at \(\theta=-\frac{\pi}{2}\). Therefore, \(\csc^{-1}(-1)=-\frac{\pi}{2}\).मान लीजिए \(\theta=\csc^{-1}(-1)\). तब \(\csc\theta=-1\), अर्थात \(\frac{1}{\sin\theta}=-1\). इसलिए \(\sin\theta=-1\)। \(\csc^{-1}x\) की principal range \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) होती है। इस range में \(\sin\theta=-1\) के लिए \(\theta=-\frac{\pi}{2}\) मिलता है। अतः \(\csc^{-1}(-1)=-\frac{\pi}{2}\).
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