Inverse Trig Functions Arcsine Arccosine and Arctangent
Arcsine Function The Arcsine function is defined as the inverse of the sine function. It is shown to the right.
Inverse Sine Graph You know that a function’s inverse is its reflection across the line y = x. Each point (x, y) on the graph of a function is matched to a point (y, x) on the graph of the inverse. The point (π, 0) is on the sine graph, and the point (0, π) is on the graph of its inverse. By this definition, the inverse sine graph would look like the blue graph to the right. (0, π) (π, 0)
Arcsine Function You will note, however, that the inverse sine graph is NOT a function. It fails the vertical line test miserably. We would like to be able to refer to a function that is the inverse of the sine function, though, so we define the arcsine function to be only a portion of the inverse sine graph. You will note that although vertical lines will pass through the inverse sine graph multiple times, each vertical line only crosses the bold section of the graph exactly once. This is the segment where the y-values lie between –π/2 to π/2. Also note that if this segment is made any longer, there will be a vertical line that crosses the segment more than once. This segment is as long as it can possibly be without failing the vertical line test.
Arcsine Domain and Range Does this segment cover all of the domain of the sine function? We noted that this segment has a domain [-1, 1] and a range [–π/2, π/2]. Thus, the inverse of this function, the sine function, must have a range [-1, 1] and a domain [–π/2, π/2] since each point (x, y) on one graph must be matched to a point (y, x) on its inverse. This means that the x- and y-values are switched between functions that are each other’s inverses. Therefore only the section of the sine function from [–π/2, π/2] has an inverse.
Arccosine and Arctangent Similarly, we must restrict the arccosine and arctangent graphs to obtain functions. The portions shown in bold are the functions f(x) = arccos(x) and g(x) = arctan(x).