Inverse Trigonometric Functions: One-Page Formula Sheet

Inverse Trigonometric Functions — formula sheet

Formulas

• tan⁻¹x + tan⁻¹y = tan⁻¹((x + y)/(1 − xy)) for xy < 1
• 2tan⁻¹x = sin⁻¹(2x/(1 + x²)) for |x| ≤ 1
• tan⁻¹x + cot⁻¹x = π/2
• sin⁻¹(−x) = −sin⁻¹x, cos⁻¹(−x) = π − cos⁻¹x

Key ideas

• sin⁻¹x has range [−π/2, π/2]; cos⁻¹x has [0, π]
• sin⁻¹x + cos⁻¹x = π/2
• sin(sin⁻¹x) = x always for x in [−1, 1]; sin⁻¹(sin x) = x only in the principal range
• Convert all terms to one function before simplifying

Traps to avoid

• Ignoring principal ranges
• Using sum formulas outside their conditions
• Sign errors for negative arguments

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