Mathematics · Ch 3 — Trigonometric Functions

Inverse Cotangent Function

3.3.6

Inverse Cotangent Function

Consider $\cot$ restricted to $(0,\pi)\to\mathbb{R}$ . Graphically this restriction is one-one and onto $\mathbb{R}$ , so its inverse exists, called the inverse cotangent function, denoted $\cot^{-1}:\mathbb{R}\to(0,\pi)$ . For $x\in\mathbb{R}$ and $\theta\in(0,\pi)$ , we write $\cot^{-1}x=\theta$ if $\cot\theta=x$ .

Ex. $\cot\dfrac{\pi}{3}=\dfrac{1}{\sqrt3}$ , where $\dfrac{1}{\sqrt3}\in\mathbb{R}$ and $\dfrac{\pi}{3}\in(0,\pi)$ , so $\cot^{-1}\left(\dfrac{1}{\sqrt3}\right)=\dfrac{\pi}{3}$ : the principal value of $\cot^{-1}\left(\dfrac{1}{\sqrt3}\right)$ is $\dfrac{\pi}{3}$ . …

Figure 3.13(a)Fig. 3.13(a) — Graph of y = cot x drawn over a wide domain, with its one-one restricted domain highlighted in bold

$Fig. 3.13(a) — Fig. 3.13(a) — Graph of y = cot x drawn over a wide domain, with its one-one restricted domain highlighted in bold$

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 $y=\cot x$ drawn only over the open interval $(0,\pi)$ on the x-axis, with vertical asymptotes as dashed lines at $x=0$ and $x=\pi$ that the curve approaches but never reaches; the curve falls steadily from $+\infty$ to $-\infty$ across the interval, one-one and onto all of $\mathbb{R} …

Figure 3.13(b)Fig. 3.13(b) — Graph of the inverse function y = cot⁻¹ x (reflection across y = x), with the principal branch in bold

$Fig. 3.13(b) — Fig. 3.13(b) — Graph of the inverse function y = cot⁻¹ x (reflection across y = x), with the principal branch in bold$

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 $y=\cot^{-1}x$ over its full domain $\mathbb{R}$ on the x-axis, obtained by reflecting the restricted cotangent graph in the line $y=x$ ; the curve falls steadily from near $\pi$ for very negative $x$ , through $\pi/2$ at the origin, down toward $0$ for very positive $x$ , always staying strictly inside $ …

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