Domain and range of the six inverse trig functions (principal value branches)छह प्रतिलोम त्रिकोणमितीय फलनों के प्रांत और परिसर (मुख्य मान शाखाएँ)Chhah inverse trig functions ke domain aur range (principal value branches)

NCERT Summary table · Class 12 Maths · Chapter 2 · interactive diagram

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Parts and what they mean

Domain (the x you may put in) (प्रांत (जो x डाल सकते हैं)) — Thick blue = allowed inputs. Arrows mean the strip goes on for ever.
Range of the principal value branch (मुख्य मान शाखा का परिसर) — Thick red = where the answer (an angle, in radians) must lie. Hollow dot = end not included.
\sin^{-1} — principal value branch (\sin^{-1} — मुख्य मान शाखा) — \sin^{-1} : [-1, 1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]
Domain of \sin^{-1}: [-1, 1] (\sin^{-1} का प्रांत: [-1, 1]) — This is the range of the restricted trig function, because domain and range swap.
Range of \sin^{-1}: [-\frac{\pi}{2}, \frac{\pi}{2}] (\sin^{-1} का परिसर: [-\frac{\pi}{2}, \frac{\pi}{2}]) — The principal value always lies here.
\cos^{-1} — principal value branch (\cos^{-1} — मुख्य मान शाखा) — \cos^{-1} : [-1, 1] \to [0, \pi]
Domain of \cos^{-1}: [-1, 1] (\cos^{-1} का प्रांत: [-1, 1]) — This is the range of the restricted trig function, because domain and range swap.
Range of \cos^{-1}: [0, \pi] (\cos^{-1} का परिसर: [0, \pi]) — The principal value always lies here.
\operatorname{cosec}^{-1} — principal value branch (\operatorname{cosec}^{-1} — मुख्य मान शाखा) — \operatorname{cosec}^{-1} : \mathbb{R} - (-1, 1) \to [-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}
Domain of \operatorname{cosec}^{-1}: \mathbb{R} - (-1, 1) (\operatorname{cosec}^{-1} का प्रांत: \mathbb{R} - (-1, 1)) — This is the range of the restricted trig function, because domain and range swap.
Range of \operatorname{cosec}^{-1}: [-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\} (\operatorname{cosec}^{-1} का परिसर: [-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}) — The principal value always lies here.
\sec^{-1} — principal value branch (\sec^{-1} — मुख्य मान शाखा) — \sec^{-1} : \mathbb{R} - (-1, 1) \to [0, \pi] - \{\frac{\pi}{2}\}
Domain of \sec^{-1}: \mathbb{R} - (-1, 1) (\sec^{-1} का प्रांत: \mathbb{R} - (-1, 1)) — This is the range of the restricted trig function, because domain and range swap.
Range of \sec^{-1}: [0, \pi] - \{\frac{\pi}{2}\} (\sec^{-1} का परिसर: [0, \pi] - \{\frac{\pi}{2}\}) — The principal value always lies here.
\tan^{-1} — principal value branch (\tan^{-1} — मुख्य मान शाखा) — \tan^{-1} : \mathbb{R} \to (-\frac{\pi}{2}, \frac{\pi}{2})
Domain of \tan^{-1}: \mathbb{R} (\tan^{-1} का प्रांत: \mathbb{R}) — This is the range of the restricted trig function, because domain and range swap.
Range of \tan^{-1}: (-\frac{\pi}{2}, \frac{\pi}{2}) (\tan^{-1} का परिसर: (-\frac{\pi}{2}, \frac{\pi}{2})) — The principal value always lies here.
\cot^{-1} — principal value branch (\cot^{-1} — मुख्य मान शाखा) — \cot^{-1} : \mathbb{R} \to (0, \pi)
Domain of \cot^{-1}: \mathbb{R} (\cot^{-1} का प्रांत: \mathbb{R}) — This is the range of the restricted trig function, because domain and range swap.
Range of \cot^{-1}: (0, \pi) (\cot^{-1} का परिसर: (0, \pi)) — The principal value always lies here.

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