Inverse Trigonometric Functions
Inverse Trigonometric Functions tests principal values, the ranges of the six inverse functions, and identity-based simplification.
Domains and ranges
sin⁻¹x: [−π/2, π/2]; cos⁻¹x: [0, π]; tan⁻¹x: (−π/2, π/2); cot⁻¹x: (0, π); sec⁻¹x: [0, π] excluding π/2; cosec⁻¹x: [−π/2, π/2] excluding 0. Every answer must be a principal value lying in these intervals.
Standard identities
sin⁻¹x + cos⁻¹x = π/2 and tan⁻¹x + cot⁻¹x = π/2. Also tan⁻¹x + tan⁻¹y = tan⁻¹((x + y)/(1 − xy)) when xy < 1; when xy > 1 add π if x + y > 0 and subtract π if x + y < 0. Further, 2tan⁻¹x = tan⁻¹(2x/(1 − x²)) for |x| < 1, tan⁻¹x + tan⁻¹(1/x) = π/2 for x > 0, and cos⁻¹(−x) = π − cos⁻¹x.
Worked example
Show tan⁻¹1 + tan⁻¹2 + tan⁻¹3 = π. First tan⁻¹2 + tan⁻¹3: here xy = 6 > 1 and x + y = 5 > 0, so the sum is π + tan⁻¹((2 + 3)/(1 − 6)) = π + tan⁻¹(−1) = π − π/4 = 3π/4. Since tan⁻¹1 = π/4, the total is π/4 + 3π/4 = π.
Traps
Adding with the simple formula when xy > 1; writing sin⁻¹(sin θ) = θ for θ outside the principal range; confusing tan⁻¹ with the reciprocal cot; ignoring the domain restriction |x| ≤ 1.