4.7 | Inverse Trigonometric Functions


For each of the six basic trigonometric functions we define an inverse trigonometric function with a suitably restricted range. The purpose of an inverse trigonometric function is to undo the computation of the corresponding trigonometric function. For example, \(\sin(\pi/6) = 1/2\). Therefore, the inverse sine of \(1/2\), denoted \(\arcsin(1/2)\) or \(\sin^{-1}(1/2)\), should return an angle whose sine value is \(1/2\) and certainly \(\pi/6\) satisfies that condition. But, it is also true that \(\sin(5\pi/6)=1/2\). So that, without defining inverse sine further, there would be at least two outputs (\(\pi/6\) and \(5\pi/6\)) for one input – contrary to the definition of a function. To prevent this phenomena from occurring we restrict the ranges of the inverse trigonometric functions. These restrictions can vary but the traditional restrictions on the ranges are listed below and the outputs of the inverse trigonometric functions, under these restrictions, are referred to as principal values. For inverse sine, the restricted range is \([-\pi/2, \pi/2]\). Thus the principal value of \(\arcsin(1/2)\) is \(\pi/6\).
Definitions of the Inverse Trigonometric Functions:
Function Domain Range (of Principal Value)
\(y = \arcsin x\) if and only if \(\sin y = x\) \(-1 \leq x \leq 1\) \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\)
\(y = \arccos x\) if and only if \(\cos y = x\) \(-1 \leq x \leq 1\) \(0 \leq y \leq \pi \)
\(y = \arctan x\) if and only if \(\tan y = x\) \(-\infty \leq x \leq \infty\) \(-\frac{\pi}{2} < y < \frac{\pi}{2}\)
You should avoid the following classical error at all costs!
$$
\require{cancel}
\require{color}
\require{enclose}
\enclose{updiagonalstrike,downdiagonalstrike}[mathcolor=”red”]{
\enclose{}[mathcolor=”black”]{
\begin{aligned}
\cos^{-1}x &= \frac{1}{\cos x}\\
\sin^{-1}x &= \frac{1}{\sin x}\\
\tan^{-1}x &= \frac{1}{\tan x}\\
\end{aligned}
}
}
$$
Inverse Properties of Trigonometric Functions:

  • If \(-1 \leq x \leq 1\) and \(-\pi/2 \leq y \leq \pi/2\), then
    $$
    \sin(\arcsin x) = x \text{ and } \arcsin(\sin y) = y.
    $$
  • If \(-1 \leq x \leq 1\) and \(0 \leq y \leq \pi\), then
    $$
    \cos(\arccos x) = x \text{ and } \arccos(\cos y) = y.
    $$
  • If \(-\infty \leq x \leq \infty\) and \(-\pi/2 < y < \pi/2\), then $$ \tan(\arctan x) = x \text{ and } \arctan(\tan y) = y. $$

E 4.7 Exercises