Inverse Trigonometry — NDA Maths

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 (7)

• cos⁻¹ and cot⁻¹ are NOT odd
  → Principal Values & Basic Properties
• tan⁻¹ is odd, but its range is OPEN
  → Principal Values & Basic Properties
• Inverse-trig answers must LAND in the principal range
  → Principal Values & Basic Properties
• sin⁻¹x + cos⁻¹x = π/2 always — don't evaluate term by term
  → Complementary Identities
• Check ab < 1 before using the sum formula
  → Sum & Difference Formulas
• Difference formula uses 1 + ab in the denominator
  → Sum & Difference Formulas
• The 2 tan⁻¹ substitutions need a validity range
  → The 2 tan⁻¹ Substitutions

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 (4)

• sin⁻¹(sin x) ≠ x outside the principal range
  → Inner-to-Outer Evaluation & sin⁻¹(sin x)
• Each cancellation uses a DIFFERENT reduction rule
  → Inner-to-Outer Evaluation & sin⁻¹(sin x)
• Double-angle tangent has 1 − tan²θ, not 1 + tan²θ
  → Double- & Half-Angle Compositions
• Convert sin⁻¹/cos⁻¹ to tan⁻¹ via the TRIANGLE, not the value
  → Converting Everything to a Tangent

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 (2)

• Reject roots that break the sum-formula validity
  → Solving Inverse-Trig Equations
• Subtended-angle denominator is d² + h₁h₂
  → Geometric Applications