Definitions of the inverse trig functions (in words):
• To find an inverse to the tangent function, we had\ve to restrict the domain of tangent to the interval $latex \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ on which tangent is one-to-one and all values of tangent are covered in this restricted domain, so that will be the range of the inverse tangent function.
Definition: $latex \tan^{-1}(x)$ or $latex \arctan(x)$ is the angle (or rotation) in the interval $latex \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ whose tangent is x.
Note: $latex \tan^{-1}(x)$ is an odd function: $latex \tan^{-1}(-x) = -\tan^{-1}(x)$
• To find an inverse to the sine function, we have to restrict the domain of sine to the interval $latex \left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ on which sine is one-to-one and all values of sine are covered in this restricted domain, so that will be the range of the inverse sine function.
Definition: $latex \sin^{-1}(x)$ or $latex \arcsin(x)$ is the angle (or rotation) in the interval $latex \left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ whose sine is x.
Note: $latex \sin^{-1}(x)$ is an odd function: $latex \sin^{-1}(-x) = -\sin^{-1}(x)$
• To find an inverse to the cosine function, we have to restrict the domain of cosine to the interval $latex \left[0,\pi\right]$ on which cosine is one-to-one and all values of cosine are covered in this restricted domain, so that will be the range of the inverse cosine function.
Definition: $latex \cos^{-1}(x)$ or $latex \arccos(x)$ is the angle (or rotation) in the interval $latex \left[0,\pi\right]$ whose cosine is x.
Note: $latex \cos{-1}(x)$ is neither even nor odd: but it does satisfy the identity $latex \cos^{-1}(-x) = \pi -\cos^{-1}(x)$
