Inverse Trigonometric Functions
Trigonometric functions are as you know periodic functions. This means the function values are repeated in regular periods and the graphs of trigonometric functions fail the horizontal line test to confirm the existence of their inverses. But often in trigonometric problems the angle or variable values are to be found when the function values are known. For this purpose, the inverse trigonometric functions are defined by restricting the domains of trigonometric functions suitably. You can see the graphs of the functions in the restricted domains and the domain and range of inverse trigonometric functions. Other than the common notation f^{1} (x) used in algebra, inverse trigonometric functions are also called arcsin, arccos arctan functions and indicated using these words. For example, the inverse sin trigonometric function is either written as sin^{1} x or arcsin x.The following tables show the graphs of trigonometric functions, the restricted domains and the domain and range for the inverse functions defined.
Graphs  Functions with restricted domains  Domain and range of inverse functions 
y = sin x Restricted Domain {x  π/2 ≤ x ≤ π/2} Range {y  1 ≤ y ≤ 1}  y = arcsin x or y = sin^{1} x Domain {x  1 ≤ x ≤ 1 Range {y  π/2 ≤ y ≤ π/2} 

y = cos x Restricted Domain {x  0 ≤ x ≤ π} Range {y  1 ≤ y ≤ 1}  y = arccos x or y = cos^{1} x Domain {x  1 ≤ x ≤ 1} Range {y  0 ≤ y ≤ π} 

y = tan x Restricted Domain {x  π/2 < x < π/2} Range {y  ∞ < y < ∞} 
y = arctan x or y = tan^{1} x Domain {x  ∞ < x < ∞} Range {y  π/2 < y < π/2} 
The restricted graphs for which the inverse is defined are shown in green color. The values in the restricted domains are called the principal values. Let us see how the inverse trigonometric graphs look like and solve few problems on them.