Mathematics
Principal value branch
प्रधान मान शाखा
In this Class 12 Mathematics topic from Inverse Trigonometric Functions, students learn why trigonometric functions must be restricted to suitable domains to have well-defined inverses. The topic explains the principal value branch, standard ranges of sin⁻¹x, cos⁻¹x, tan⁻¹x and related functions, and how these ranges determine the unique value returned by an inverse function. Students also practise interpreting graphs, selecting correct branches, and applying principal-value conventions while simplifying expressions and solving problems.
Practice questions
If \(\theta=\cos^{-1}\left(\frac{12}{13}\right)\), what is the value of \(\sin\theta\)?EasyLevel 32What is the value of \(\sin^{-1}\left(\sin\left(-\frac{\pi}{4}\right)\right)\)?EasyLevel 32What is the value of \(\cos^{-1}\left(\cos 0\right)\)?EasyLevel 32What is cos⁻¹(−x) equal to when x ∈ [−1, 1]?MediumLevel 33What is the principal value of \(\tan^{-1}(\tan\frac{3\pi}{4})\)?MediumLevel 31If \(\alpha=\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\), what is \(\alpha\) equal to?MediumLevel 31What is the value of \(\cos^{-1}\left(\frac{\sqrt{2}}{2}\right)\)?MediumLevel 31What is the principal value of \(\sin^{-1}\left(\sin\left(-\frac{5\pi}{6}\right)\right)\)?MediumLevel 31For a real number \(x\), in which interval does the principal value of \(\tan^{-1}x\) lie?MediumLevel 31What is the principal value of \(\sin^{-1}\left(\sin\frac{7\pi}{6}\right)\)?MediumLevel 31What is the principal value of \(\cos^{-1}\left(\cos\left(-\frac{\pi}{4}\right)\right)\)?MediumLevel 31What is the principal value of \(cosec^{-1}\left(\frac{2}{\sqrt{3}}\right)\)?MediumLevel 31What is the principal value of \(\sin^{-1}\left(\frac{1}{2}\right)\)?MediumLevel 32When is \(\sin^{-1}(\sin\theta)=\theta\) definitely true?MediumLevel 32When is \(\cos^{-1}(\cos\theta)=\theta\) definitely true?MediumLevel 32When is \(\tan^{-1}(\tan\theta)=\theta\) definitely true?MediumLevel 32What is the principal value of \(cosec^{-1}2\)?MediumLevel 32What is the principal value of \(\cos^{-1}\left(\cos\frac{4\pi}{3}\right)\)?MediumLevel 32If \(\tan^{-1}x=-\frac{\pi}{3}\), what is the value of \(x\)?MediumLevel 32What is the principal value of \(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)?MediumLevel 32