Knowledge organisers

Inverse Trigonometric Functions: CBSE Class 12 Maths knowledge organiser

Everything to know about inverse trigonometric functions on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Principal value branch: The range chosen so that an inverse trigonometric function gives exactly one value.
Inverse sine: sin⁻¹x is the angle in [−π/2, π/2] whose sine is x.
Domain restriction: Trig functions are made one-one by restricting their domains before inverting.

Key formulas

Principal ranges	\(\sin^{-1}:[-1,1]\to\left[-\tfrac\pi2,\tfrac\pi2\right],\) \(\cos^{-1}:[-1,1]\to[0,\pi],\) \(\tan^{-1}:\mathbb R\to\left(-\tfrac\pi2,\tfrac\pi2\right)\)
More ranges	\(\cot^{-1}:\mathbb R\to(0,\pi),\) \(\sec^{-1}:[0,\pi]-\{\tfrac\pi2\},\) \(\cosec^{-1}:\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}\)
\(\sec^{-1}\), \(\cosec^{-1}\) domain	\(\mathbb R-(-1,1)\)
\(\sin(\sin^{-1}x)=x\) for \(x\in[-1,1]\); \(\sin^{-1}(\sin x)=x\) only for \(x\in[-\tfrac\pi2,\tfrac\pi2]\) (similarly for the others)

Worked example

\(\sin^{-1}\left(-\dfrac12\right) = -\dfrac{\pi}{6}\), \(\cos^{-1}\left(-\dfrac12\right) = \dfrac{2\pi}{3}\).

Common mistakes

Writing \(\sin^{-1}\left(\sin\dfrac{2\pi}{3}\right) = \dfrac{2\pi}{3}\) — the answer must lie in the principal range.
Giving \(\cos^{-1}(-x)\) a negative value: \(\cos^{-1}\) is never negative.
Mixing up ranges of \(\cot^{-1}\) \((0, \pi)\) and \(\tan^{-1}\) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Ignoring the sign of a square root after a substitution (e.g. \(\sqrt{\cos^2\theta} = |\cos\theta|\)).

You should be able to…

Write the principal value branch of every inverse trig function and find principal values of standard values.
Find domains, simplify compositions like sin⁻¹(sin x) using the principal range and evaluate expressions such as cos(tan⁻¹ ¾).
Write complete multi-part answers (the case studies, the 5-mark simplification and the multi-part domain questions), stating the branch you use at each step.
Handle radian inputs like sin⁻¹(sin 3), substitution-based simplifications and justifying the signs when you write sin θ and cos θ in terms of x.

The printable sheet

Inverse Trigonometric Functions knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Inverse Trigonometric Functions knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers