# Relationships between Inverse Trigonometric Functions

Real number (argument of an inverse function): $$x$$
CInverse 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. $${\mathop{\rm arccot}\nolimits}\,x =$$ $$\pi + \arctan {\large\frac{1}{x}\normalsize},\;$$ $$x \lt 0$$