Inverse Trig Functions 1

Review of Trig Functions

Before discussing inverse trig functions we will recall how Sine, Cosine, and Tangent are defined. All three of them take, as input, an angle. Measuring this angle from the positive x-axis, we can define a ray, one directional line emanating from the origin and creating the angle that was given as input. This ray will intersect the unit circle at one point. From there Sine, Cosine, and Tangent, output some information related to this point on the unit circle and the ray. Sine outputs the x-coordinate of the point, Cosine outputs the y coordinate, and Tangent outputs the ratio of the two, dividing the x-coordinate by the y-coordinate.

Inverse Sine

As we trace around the unit circle, we can observe that the y-coordinate will go from 0, to 1, back to 0, to -1, and finally completing the rotation with a y-coordinate of 0. Notice that every number occurs twice during each complete rotation, a picture helps. In order to define an inverse we have to restrict the domain, this will allow the restricted graph to pass the horizontal line test. Namely, we restrict the angles to $[-{\frac {\pi }{2}},{\frac {\pi }{2}}]$. Now inverse Sine will take as input a y-coordinate, associated to one of the angles in the restricted domain, and output the angle.

Inverse Cosine

Inverse Cosine behaves vary similarly to inverse sine, except Cosine deals with x-coordinates, and has we restrict ourselves to the top half of the unit circle, so the only angles we care about are $[0,\pi ]$

Inverse Trig

Inverse Trig takes, as input, the y-coordinate divided by the x-coordinate. So it is undefined when the angle is ${\frac {\pi }{2}}{\text{ or }}{\frac {3\pi }{2}}$ We restrict the domain of the Tangent function to $(-{\frac {\pi }{2}},{\frac {\pi }{2}})$

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