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A student claims that \(\tan^{-1}(\tan(3\pi/4))=3\pi/4\). What is the correct evaluation of this claim?एक विद्यार्थी कहता है कि \(\tan^{-1}(\tan(3\pi/4))=3\pi/4\)। इस कथन का सही मूल्यांकन क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: The claim is incorrect; the correct value is \(-\pi/4\), because the principal value of \(\tan^{-1}\) lies in \((-\pi/2,\pi/2)\).कथन गलत है; सही मान \(-\pi/4\) है, क्योंकि \(\tan^{-1}\) का प्रमुख मान \((-\pi/2,\pi/2)\) में होता है।
\(\tan(3\pi/4)=-1\). Since the principal-value range of \(\tan^{-1}\) is \((-\pi/2,\pi/2)\), \(\tan^{-1}(-1)=-\pi/4\), not \(3\pi/4\). Exam tip: always check the principal-value range before cancelling inverse functions.\(\tan(3\pi/4)=-1\)। चूँकि \(\tan^{-1}\) का प्रमुख मान \((-\pi/2,\pi/2)\) में लिया जाता है, इसलिए \(\tan^{-1}(-1)=-\pi/4\), न कि \(3\pi/4\)। परीक्षा में पहले प्रमुख मान की सीमा अवश्य जाँचें।
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