NDA Maths · Teaching notes

Inverse Trigonometry — NDA Maths

Inverse trigonometry asks the reverse question — given a ratio, which angle produced it? — with one twist: the answer must lie in a fixed principal-value range. 34 PYQs span 2017–2026, formula-heavy and unforgiving on the range. The notes teach in three movements, foundations first: (1) Identities, Properties & Sum-Difference — the principal-value branches, the odd/even rules, the complementary identities (sin⁻¹x + cos⁻¹x = π/2), and the tan⁻¹a ± tan⁻¹b sum formula with its 2 tan⁻¹ substitutions; (2) Evaluation of Composite Expressions — reducing sin⁻¹(sin x) to the principal value, peeling nested compositions from the inside out, and the double/half-angle compositions; (3) Solving Equations & Geometric Applications — solving inverse-trig equations via the complementary identity (watching the validity of the sum formula), and angle-of-elevation problems. Fix the principal range first; every clean answer depends on it. Every PYQ is tagged.

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Subtopic notes

PYQ weightage by concept

9 concepts · 34 PYQs — where the marks actually sit, so you know what to drill first

Identities, Properties & Sum-Difference Formulas17 PYQs · 50%

Concept	PYQs	Share
Sum & Difference Formulas	7	21%
Complementary Identities	6	18%
Principal Values & Basic Properties	3	9%
The 2 tan⁻¹ Substitutions	1	3%

Evaluating Composite Inverse Expressions11 PYQs · 32%

Concept	PYQs	Share
Double- & Half-Angle Compositions	6	18%
Inner-to-Outer Evaluation & sin⁻¹(sin x)	3	9%
Converting Everything to a Tangent	2	6%

Solving Equations & Geometric Applications6 PYQs · 18%

Concept	PYQs	Share
Solving Inverse-Trig Equations	4	12%
Geometric Applications	2	6%

Formula & revision sheet

9 formulas · 4 gotchas across all subtopics — the exam-eve cheat-sheet

Identities, Properties & Sum-Difference Formulas

Formulas (4)

Principal Values & Basic Properties · Principal ranges $\sin^{-1}x \in [-\tfrac{\pi}{2},\tfrac{\pi}{2}], \quad \cos^{-1}x \in [0,\pi], \quad \tan^{-1}x \in (-\tfrac{\pi}{2},\tfrac{\pi}{2})$
Complementary Identities · Complementary pairs $\sin^{-1}x + \cos^{-1}x = \tfrac{\pi}{2}, \quad \tan^{-1}x + \cot^{-1}x = \tfrac{\pi}{2}$
Sum & Difference Formulas · Arctangent sum $\tan^{-1}a + \tan^{-1}b = \tan^{-1}\dfrac{a+b}{1-ab}\quad (ab<1)$
The 2 tan⁻¹ Substitutions · Double-angle substitution $\tan^{-1}\dfrac{2x}{1-x^2} = 2\tan^{-1}x$

Watch out for (2)

cos⁻¹ and cot⁻¹ are NOT odd
→ Principal Values & Basic Properties
Check ab < 1 before using the sum formula
→ Sum & Difference Formulas

Evaluating Composite Inverse Expressions

Formulas (3)

Inner-to-Outer Evaluation & sin⁻¹(sin x) · Principal-range reduction $\sin^{-1}(\sin x) = x \ \text{ only if } x \in \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$
Double- & Half-Angle Compositions · Double-angle tangent $\tan(2\tan^{-1}x) = \dfrac{2x}{1 - x^2}$
Converting Everything to a Tangent · Triangle → tangent $\sin^{-1}\tfrac{3}{5} = \tan^{-1}\tfrac{3}{4}, \quad \cot^{-1}\tfrac{3}{2} = \tan^{-1}\tfrac{2}{3}$

Watch out for (1)

sin⁻¹(sin x) ≠ x outside the principal range
→ Inner-to-Outer Evaluation & sin⁻¹(sin x)

Solving Equations & Geometric Applications

Formulas (2)

Solving Inverse-Trig Equations · Collapse with the complementary identity $a\sin^{-1}x + b\cos^{-1}x = c \ \xrightarrow{\cos^{-1}x = \frac{\pi}{2}-\sin^{-1}x}\ \text{one unknown}$
Geometric Applications · Subtended angle $\tan^{-1}\dfrac{h_2}{d} - \tan^{-1}\dfrac{h_1}{d} = \tan^{-1}\dfrac{(h_2-h_1)\,d}{d^2 + h_1 h_2}$

Watch out for (1)

Reject roots that break the sum-formula validity
→ Solving Inverse-Trig Equations