5.5 Inverse Trigonometric Functions and Their Graphs
Ideas in this lesson
Restricted inverse definitions
Exact and numerical inverse evaluation
Cancellation with domains
Mixed trig-inverse compositions
Inverse identity conjectures and proof
Compare compositions globally
Inverse reciprocal branches
Restricted inverse definitions · Worked example
Explain why the full tan function has no inverse function, and why restricting its input to \((-\dfrac\pi2,\dfrac\pi2)\) works. Read the graph of the restricted function (blue), inverse (pink), and reflection line (green). State the inverse domain/range, included endpoints or horizontal asymptotes, and evaluate \(\arctan(2)\) to five decimals. Distinguish inverse notation from a reciprocal. Show the requested reasoning; written work and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: Domain \(\mathbb{R}\); range \((-\dfrac\pi2,\dfrac\pi2)\); value \(1.10715\).
The unrestricted function repeats outputs, so it fails the horizontal line test. On \((-\dfrac\pi2,\dfrac\pi2)\) it is strictly increasing and takes every output of its original range once. Reflection swaps input and output: \(\arctan(u)=w\quad\Longleftrightarrow\quad\tan(w)=u\) with \(w\in(-\dfrac\pi2,\dfrac\pi2)\). Hence the inverse domain is \(\mathbb{R}\) and its range is \((-\dfrac\pi2,\dfrac\pi2)\). The finite endpoints are excluded. The horizontal asymptotes are \(y=-\dfrac\pi2\) as input tends to negative infinity and \(y=\dfrac\pi2\) as input tends to positive infinity. \(\arctan(2)\approx 1.10715\) in radians. \(\tan^{-1}(u)\) denotes this inverse here, not \(\dfrac1{\tan u}\). Written-work review:
Use the stated definitions and the complete allowed domain.
Show all intermediate values and justify signs, restrictions and conclusions.
Give every requested result in the stated order and units.