Cheat Sheet – Inverse Trigonometric Functions PDF

DOUBLEROOT Cheat Sheet – Inverse Trigonometric Functions x+y Domain/Range 32. tan-1x + tan-1y = tan-1...

DOUBLEROOT Cheat Sheet – Inverse Trigonometric Functions x+y Domain/Range 32. tan-1x + tan-1y = tan-1 1 – xy Function Domain Range x > 0, y > 0, xy < 1 sin-1x [–1, 1] [–π/2, π/2] x+y cos-1x [–1, 1] [0, π] 33. tan-1x + tan-1y = π + tan-1 1 – xy tan-1x (–∞, ∞) (–π/2, π/2) x > 0, y > 0, xy > 1 cot-1x (–∞, ∞) (0, π) 2x 34. 2tan-1x = sin-1 x ϵ [–1, 1] sec-1x R – (–1,1) [0, π] – {π/2} 1+x2 2x csc-1x R – (–1,1) [–π/2, π/2] – {0} 35. 2tan-1x = π – sin-1 x ϵ (1, ∞) 1+x2 2x Identities 36. 2tan-1x = – π – sin -1 x ϵ (–∞, –1) 1+x2 1. sin(sin-1x) = x x ϵ [–1, 1] 1−x2 37. 2tan-1x = cos-1 x ϵ [0, ∞) 2. cos(cos-1x) = x x ϵ [–1, 1] 1+x2 3. tan(tan-1x) = x xϵR 1−x2 38. 2tan-1x = – cos -1 x ϵ (–∞, 0) 4. cot(cot-1x) = x xϵR 1+x2 2x 5. sec(sec-1x) = x x ϵ R – (–1, 1) 39. 2tan-1x = tan-1 x ϵ (–1, 1) 1−x2 6. csc(csc-1x) = x x ϵ R – (–1, 1) 2x 40. 2tan-1x = π + tan -1 x ϵ (1, ∞) 1−x2 7. sin-1(sinx) = x x ϵ [–π/2, π/2] 2x 41. 2tan-1x = – π + tan -1 x ϵ (–∞,– 1) 8. cos-1(cosx) = x x ϵ [0, π] 1−x2 9. tan-1(tanx) = x x ϵ (–π/2, π/2) 10. cot-1(cotx) = x x ϵ (0, π) 42. 2sin-1x = sin-1(2x√1 − x 2 ) x ϵ [–1/√2, 1/√2] 11. sec-1(secx) = x x ϵ [0, π] – {π/2} 43. 2sin-1x = π – sin-1(2x√1 − x 2 ) x ϵ (1/√2, 1] 12. csc-1(cscx) = x x ϵ [–π/2, π/2] – {0} 44. 2sin-1x = – π – sin-1(2x√1 − x 2 ) x ϵ [– 1, –1/√2) 13. sin-1x + cos-1x = π/2 x ϵ [–1, 1] 45. 2cos-1x = cos-1(2x2 – 1) x ϵ [0, 1] 14. tan-1x + cot-1x = π/2 xϵR 46. 2cos-1x = 2π – cos-1(2x2 – 1) x ϵ [–1, 0) 15. sec-1x + csc-1x = π/2 x ϵ R – (–1, 1) 47. 3sin-1x = sin-1(3x – 4x3) x ϵ [–1/2, 1/2] 16. sin (– x) = – sin x -1 -1 x ϵ [–1, 1] 48. 3sin-1x = π – sin-1(3x – 4x3) x ϵ (1/2, 1] 17. cos-1(– x) = π – cos-1x x ϵ [–1, 1] 49. 3sin-1x = – π – sin-1(3x – 4x3) x ϵ [–1, –1/2) 18. tan-1(– x) = – tan-1x xϵR 19. cot-1(– x) = π – cot-1x xϵR 50. 3cos-1x = cos-1(4x3 – 3x) x ϵ [1/2, 1] 20. sec-1(– x) = π – sec-1x x ϵ R – (–1, 1) 51. 3cos-1x = 2π – cos-1(4x3 – 3x) x ϵ [–1/2, 1/2) 21. csc-1(– x) = – csc-1x x ϵ R – (–1, 1) 52. 3cos-1x = 2π + cos-1(4x3 – 3x) x ϵ [–1, –1/2) 3x–x3 53. 3tan-1x = tan-1 x ϵ (–1/√3, 1/√3) 22. sin x = csc (1/x) -1 -1 x ϵ [–1, 1] – {0} 1–3x2 23. csc-1x = sin-1(1/x) x ϵ R – (–1, 1) 3x–x3 24. cos-1x = sec-1(1/x) x ϵ [–1, 1] – {0} 54. 3tan-1x = π + tan -1 1–3x2 x ϵ (1/√3, ∞) 25. sec-1x = cos-1(1/x) x ϵ R – (–1, 1) 3x–x3 55. 3tan-1x = – π + tan-1 x ϵ (–∞, – 1/√3) 26. tan-1x = cot-1(1/x) x ϵ (0, ∞) 1–3x2 tan-1x = cot-1(1/x) – π x ϵ (–∞, 0) 27. cot-1x = tan-1(1/x) x ϵ (0, ∞) 56. sin-1x + sin-1y = sin-1 x 1 − y 2 + y√1 − x 2 cot-1x = tan-1(1/x) + π x ϵ (–∞, 0) x2 + y2 ≤ 1, or xy ≤ 0 57. sin-1x + sin-1y = – sin-1 x 1 − y 2 + y√1 − x 2 + π 28. sin-1x = cos-1√1 − x 2 x ϵ [0, 1] x2 + y2 > 1, x, y ϵ (0, 1] 29. sin-1x = – cos-1√1 − x 2 x ϵ [-1, 0) 58. sin-1x + sin-1y = – sin-1 x 1 − y 2 + y√1 − x 2 – π 30. cos-1x = sin-1√1 − x 2 x ϵ [0, 1] x2 + y2 > 1, x, y ϵ [–1, 0) 31. cos-1x = π – sin-1√1 − x 2 x ϵ [-1, 0) © www.doubleroot.in