Inverse Trigonometric Functions and their Graphs:
A function has an inverse function strictly when no horizontal line intersects the graph more than once. Since trig. functions don't have inverse functions, and it is useful to have an inverse function, a restriction is applied to the domain so that an inverse function might exist. Certain conditions must be met in order for the trig. functions to have an inverse function.
  1. Each value of the range is only taken on once so it can pass the horizontal line test.
  2. The range of the function with its restricted domain is the same as the range of the original function.
  3. The domain includes the most commonly used numbers (or angles)
     0 < x < p/2.
  4. The graph is connected (if possible).
Â
Inverse sine:
If -1 < x < 1, then f(x) = sin-1 x ( or f(x) = arcsin x), if and only if sin f(x) = x and - p/2 < f(x) < p/2.
Inverse cosine:
If -1 < x < 1, then f(x) = cos-1x ( or f(x) = arccos x), if and only if cos f(x) = x and
0 < f(x) < p.
Inverse tangent:
If x is any real number, then f(x) = tan-1 x ( or f(x) = arctan x), if and
only if tan f(x) = x and - p/2 < f(x) < p/2.
Inverse cosecant:
If | x | ³ 1, then f(x) = csc-1 x ( or f(x) = arccsc x), if and only if
csc f(x) = x and - p/2 < f(x) < p/2 , y ¹ 0.
Inverse secant:
If | x | ³ 1, then f(x) = sec-1 x ( or f(x) = arcsec x), if and only if
sec f(x) = x and 0 < f(x) < p, y ¹ p/2.
Inverse cotangent:
If x is any real number, then f(x) = cot-1 x ( or f(x) = arccot x), if and only if cot f(x) = x and 0 < f(x)< p.
