TrigonometryHigh Yield (1-2 Qs in JEE)

Inverse Trigonometric Functions (ITF) Formulas

Principal branches, domain & range, complementary angle relations, and sum of inverse tangents.

8 Formulas•470+ Questions in Bank•

Concept Diagram

Trigonometric Unit Circle

Standard angles (0, π/6, π/4, π/3, π/2), (cos θ, sin θ) coordinates, and ASTC quadrant sign rules.

\sin^{-1}x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right], \quad \cos^{-1}x \in [0, \pi], \quad \tan^{-1}x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

Standard principal branches defined to make trigonometric functions bijective.

Examiner Tip: Always reduce angles outside principal branch into the fundamental domain first.

\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}, \quad \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \quad (x \in \text{Domain})

Sum of co-functions equals π/2 everywhere in their common domain.

Examiner Tip: sec⁻¹x + csc⁻¹x = π/2 for |x| ≥ 1.

\tan^{-1}x + \tan^{-1}y = \begin{cases} \tan^{-1}\left(\frac{x+y}{1-xy}\right) & xy < 1 \\ \pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right) & x,y > 0, \; xy > 1 \end{cases}

Combines two inverse tangents into one.

Examiner Tip: Verify the sign of xy before applying the formula without π adjustment.

2\tan^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \tan^{-1}\left(\frac{2x}{1-x^2}\right)

Interconversion between 2 tan⁻¹x and other inverse functions.

Examiner Tip: Check domain conditions: cos⁻¹((1-x²)/(1+x²)) = 2 tan⁻¹x holds for x ≥ 0.

\sin^{-1}(-x) = -\sin^{-1}x, \quad \cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x

Odd vs supplementary behavior for negative arguments.

Examiner Tip: sec⁻¹(-x) = π - sec⁻¹x and cot⁻¹(-x) = π - cot⁻¹x.

\sum_{r=1}^n \tan^{-1}\left(\frac{x_r - x_{r-1}}{1 + x_r x_{r-1}}\right) = \tan^{-1}x_n - \tan^{-1}x_0, \quad \tan^{-1}\left(\frac{1}{1 + r + r^2}\right) = \tan^{-1}(r+1) - \tan^{-1}r

Decomposition of rational algebraic arguments into differences of consecutive inverse tangents for infinite summation.

Examiner Tip: For ∑_{r=1}^∞ tan⁻¹(1/(r² + r + 1)): Sum = lim_{n→∞} (tan⁻¹(n+1) - tan⁻¹ 1) = π/2 - π/4 = π/4.

\sin^{-1}x \pm \sin^{-1}y = \sin^{-1}\left(x\sqrt{1-y^2} \pm y\sqrt{1-x^2}\right), \quad \cos^{-1}x \pm \cos^{-1}y = \cos^{-1}\left(xy \mp \sqrt{1-x^2}\sqrt{1-y^2}\right)

Standard addition and subtraction formulas for inverse sines and cosines within primary domains.

Examiner Tip: For sin⁻¹x + sin⁻¹y: if x² + y² > 1 and x,y > 0, the sum exceeds π/2 and equals π - sin⁻¹(x√(1-y²) + y√(1-x²)).

(\sin^{-1}x)^2 + (\cos^{-1}x)^2 \in \left[\frac{\pi^2}{8}, \frac{5\pi^2}{4}\right], \quad (\sin^{-1}x)^3 + (\cos^{-1}x)^3 \in \left[\frac{\pi^3}{32}, \frac{7\pi^3}{8}\right]

Ranges of quadratic and cubic symmetric combinations of inverse sine and cosine.

Examiner Tip: Minimum occurs when sin⁻¹x = cos⁻¹x = π/4 (i.e. x = 1/√2). Maximum occurs at x = -1.

Frequently tested misconceptions in Inverse Trigonometric Functions (ITF)