Formulas and Tables

Trigonometry

Relationships between Inverse Trigonometric Functions

Real number (argument of an inverse function): \(x\)

Inverse trigonometric functions: \(\arcsin x,\) \(\arccos x,\) \(\arctan x,\) \(\text {arccot }x\)

  1. Arcsine of a negative number
    \(\arcsin \left( { – x} \right) =\) \( – \arcsin x\)
  2. Expressing the arcsine in terms of arccosine
    \(\arcsin x =\) \( -\pi/2 – \arccos x\)
  3. \(\arcsin x =\) \( \arccos \sqrt {1 – {x^2}} ,\;\) \(0 \le x \le 1\)
  4. \(\arcsin x =\) \( -\arccos \sqrt {1 – {x^2}} ,\;\) \(-1 \le x \le 0\)
  5. \(\arcsin x =\) \( \arctan {\large\frac{x}{{\sqrt {1 – {x^2}} }}\normalsize},\;\) \({x^2} \le 1\)
  6. \(\arcsin x =\) \({\mathop{\rm arccot}\nolimits}\,{\large\frac{{\sqrt {1 – {x^2}} }}{x}\normalsize},\;\) \(0 \lt x \le 1\)
  7. \(\arcsin x =\) \({\mathop{\rm arccot}\nolimits}\,{\large\frac{{\sqrt {1 – {x^2}} }}{x}\normalsize} – \pi,\;\) \(-1 \le x \lt 0\)
  8. Arccosine of a negative number
    \(\arccos \left( { – x} \right) =\) \( \pi – \arccos x\)
  9. Expressing the arccosine in terms of arcsine
    \(\arccos x =\) \( \pi/2 – \arcsin x\)
  10. \(\arccos x =\) \( \arcsin \sqrt {1 – {x^2}} ,\;\) \(0 \le x \le 1\)
  11. \(\arccos x =\) \(\pi – \arcsin \sqrt {1 – {x^2}} ,\;\) \(-1 \le x \le 0\)
  12. \(\arccos x =\) \( \arctan \large\frac{{\sqrt {1 – {x^2}} }}{x}\normalsize,\;\) \(0 \lt x \le 1\)
  13. \(\arccos x =\) \( \pi + \arctan \large\frac{{\sqrt {1 – {x^2}} }}{x}\normalsize,\;\) \(-1 \le x \lt 0\)
  14. \(\arccos x =\) \({\mathop{\rm arccot}\nolimits}\,\large\frac{x}{{\sqrt {1 – {x^2}} }}\normalsize,\;\) \(- 1 \lt x \lt 1\)
  15. Arctangent of a negative number
    \(\arctan \left( { – x} \right) =\) \( – \arctan x\)
  16. Expressing the arctangent in terms of arccotangent
    \(\arctan x =\) \( \pi/2 – \text {arccot }x\)
  17. \(\arctan x = \arcsin \large\frac{x}{{\sqrt {1 + {x^2}} }}\normalsize\)
  18. \(\arctan x =\) \( \arccos \large\frac{1}{{\sqrt {1 + {x^2}} }}\normalsize,\;\) \(x \ge 0\)
  19. \(\arctan x =\) \( -\arccos \large\frac{1}{{\sqrt {1 + {x^2}} }}\normalsize,\;\) \(x \le 0\)
  20. \(\arctan x =\) \( {\large\frac{\pi }{2}\normalsize} – \arctan {\large\frac{1}{x}\normalsize},\;\) \(x \gt 0\)
  21. \(\arctan x =\) \( -{\large\frac{\pi }{2}\normalsize} – \arctan {\large\frac{1}{x}\normalsize},\;\) \(x \lt 0\)
  22. \(\arctan x =\) \( {\mathop{\rm arccot}\nolimits}\,{\large\frac{1}{x}\normalsize},\;\) \(x \gt 0\)
  23. \(\arctan x =\) \( {\mathop{\rm arccot}\nolimits}\,{\large\frac{1}{x}\normalsize} – \pi,\;\) \(x \lt 0\)
  24. Arccotangent of a negative number
    \({\mathop{\rm arccot}\nolimits} \left( { – x} \right) = \pi – {\mathop{\rm arccot}\nolimits}\,x\)
  25. Expressing the arccotangent in terms of arctangent
    \({\mathop{\rm arccot}\nolimits}\,x =\) \( \pi/2 – \arctan x\)
  26. \({\mathop{\rm arccot}\nolimits}\,x =\) \( \arcsin {\large\frac{1}{{\sqrt {1 + {x^2}} }}\normalsize},\;\) \(x \gt 0\)
  27. \({\mathop{\rm arccot}\nolimits}\,x =\) \(\pi – \arcsin {\large\frac{1}{{\sqrt {1 + {x^2}} }}\normalsize},\;\) \(x \lt 0\)
  28. \({\mathop{\rm arccot}\nolimits}\,x =\) \( \arccos {\large\frac{x}{{\sqrt {1 + {x^2}} }}\normalsize}\)
  29. \({\mathop{\rm arccot}\nolimits}\,x =\) \( \arctan {\large\frac{1}{x}\normalsize},\;\) \(x \gt 0\)
  30. \({\mathop{\rm arccot}\nolimits}\,x =\) \(\pi + \arctan {\large\frac{1}{x}\normalsize},\;\) \(x \lt 0\)