The inverse trig functions arcsin, arccos and arctan, including their principal-value domains and ranges and the graphs as reflections of the restricted trig functions in y=x.
The inverse of sin is arcsin, also written sin−1. Its domain is the range of sin: (−1,1), and its range is the domain of sinx, restricted so that the inverse function passes the vertical line test: [−2π,2π].
The graph of y=arcsinx is the mirror image of sinx (restricted to −2π<x<2π) in the line y=x, giving an increasing function:
The inverse of cos is arccos, also written cos−1. Its domain is the range of cos: (−1,1), and its range is the domain of cosx, restricted so that the inverse function passes the vertical line test: [0,π].
The graph of y=arccosx is the mirror image of cosx (restricted to 0<x<π) in the line y=x, giving a decreasing function:
The inverse of tan is arctan, also written tan−1. Its domain is the range of tan: all real numbers, and its range is the domain of tanx, restricted so that the inverse function passes the vertical line test: (−2π,2π).
The graph of y=arctanx is the mirror image of sinx (restricted to −2π<x<2π) in the line y=x, giving an increasing function:
Nice work completing Inverse trig functions, here's a quick recap of what we covered:
Exercises checked off
The inverse trig functions arcsin, arccos and arctan, including their principal-value domains and ranges and the graphs as reflections of the restricted trig functions in y=x.
The inverse of sin is arcsin, also written sin−1. Its domain is the range of sin: (−1,1), and its range is the domain of sinx, restricted so that the inverse function passes the vertical line test: [−2π,2π].
The graph of y=arcsinx is the mirror image of sinx (restricted to −2π<x<2π) in the line y=x, giving an increasing function:
The inverse of cos is arccos, also written cos−1. Its domain is the range of cos: (−1,1), and its range is the domain of cosx, restricted so that the inverse function passes the vertical line test: [0,π].
The graph of y=arccosx is the mirror image of cosx (restricted to 0<x<π) in the line y=x, giving a decreasing function:
The inverse of tan is arctan, also written tan−1. Its domain is the range of tan: all real numbers, and its range is the domain of tanx, restricted so that the inverse function passes the vertical line test: (−2π,2π).
The graph of y=arctanx is the mirror image of sinx (restricted to −2π<x<2π) in the line y=x, giving an increasing function:
Nice work completing Inverse trig functions, here's a quick recap of what we covered:
Exercises checked off