Mathematics · Ch 3 — Trigonometric Functions

Inverse Tangent Function

3.3.3

Inverse Tangent Function

Consider $\tan$ restricted to $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\to\mathbb{R}$ . Graphically this restriction is one-one and onto $\mathbb{R}$ , so its inverse exists, called the inverse tangent function, denoted $\tan^{-1}:\mathbb{R}\to\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ . For $x\in\mathbb{R}$ and $\theta\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ , we write $\tan^{-1}x=\theta$ if $\tan\theta=x$ .

Ex. $\tan\dfrac{\pi}{4}=1$ , where $1\in\mathbb{R}$ and $\dfrac{\pi}{4}\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ , so $\tan^{-1}(1)=\dfrac{\pi}{4}$ : the principal value of $\tan^{-1}1$ is $\dfrac{\pi}{4}$ . …

Figure 3.10(a)Fig. 3.10(a) — Graph of y = tan x drawn over a wide domain, with its one-one restricted domain highlighted in bold

$Fig. 3.10(a) — Fig. 3.10(a) — Graph of y = tan x drawn over a wide domain, with its one-one restricted domain highlighted in bold$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Shows the graph of $y=\tan x$ drawn only over the open restricted interval $\left(-\pi/2,\pi/2\right)$ on the x-axis, with vertical asymptotes marked as dashed lines at $x=\pm\pi/2$ that the curve approaches but never reaches; the curve rises without bound between these asymptotes, one-one on this interval and onto all of $\mat …

Figure 3.10(b)Fig. 3.10(b) — Graph of the inverse function y = tan⁻¹ x (reflection across y = x), with the principal branch in bold

$Fig. 3.10(b) — Fig. 3.10(b) — Graph of the inverse function y = tan⁻¹ x (reflection across y = x), with the principal branch in bold$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Shows the graph of $y=\tan^{-1}x$ over its full domain $\mathbb{R}$ on the x-axis, obtained by reflecting the restricted tangent graph in the line $y=x$ ; the curve rises slowly from below, passing through the origin, and flattens out approaching but never reaching the horizontal asymptotes $y=\pm\pi/2$ , confirming the st …

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