Basic Concepts

What Fig. 2.1 Shows

The figure is arranged as three separate coordinate panels that together tell the story of how the inverse sine function is constructed from the sine function.

Panel (i) — top centre — shows the familiar graph of $y = \sin x$. The $x$-axis is marked at intervals of $\pi/2$ from $-5\pi/2$ to $5\pi/2$, and the $y$-axis shows the range $[-1, 1]$. The curve oscillates as a smooth wave between $y = 1$ and $y = -1$. A thicker, darker segment of this curve is highlighted over the interval $[-\pi/2, \pi/2]$ on the $x$-axis. This dark portion is the principal-value branch of the sine function — the restricted domain on which sine becomes one-to-one.

Panel (ii) — bottom left — shows $y = \sin^{-1} x$. This graph is obtained by reflecting the sine curve across the line $y = x$. The domain is now $[-1, 1]$ on the $x$-axis, and the $y$-axis runs through multiples of $\pi/2$ up to $\pm 5\pi/2$. The curve rises vertically from $(-1, -\pi/2)$ to $(1, \pi/2)$, then continues in repeating vertical segments. The dark portion of this curve lies between $y = -\pi/2$ and $y = \pi/2$ — this is the principal value branch of $\sin^{-1}$.

Panel (iii) — bottom right — superimposes three graphs on the same axes: $y = \sin x$, $y = \sin^{-1} x$, and the straight line $y = x$ (drawn with arrows at both ends). The two curves are mirror images of each other across this line. Each curve has its principal branch darkened, making the reflection relationship visually immediate.

The Core Idea

The figure teaches a fundamental principle: the graph of an inverse function is the reflection of the original function's graph across the line $y = x$. This works because if $(a, b)$ lies on the graph of $f$, then $(b, a)$ lies on the graph of $f^{-1}$. Swapping coordinates is equivalent to reflecting across $y = x$.

But sine is not one-to-one on its entire domain $\mathbb{R}$. To define an inverse, we must restrict sine to an interval where it is one-to-one and onto $[-1, 1]$. The standard choice is $[-\pi/2, \pi/2]$, shown as the dark segment in panel (i). This restricted sine is then reflected to give the principal branch of $\sin^{-1}$ in panel (ii), with range $[-\pi/2, \pi/2]$.

Key Formulas Illustrated

The figure directly supports these defining relationships:

From this, two composition identities follow:

The second identity holds only when $x$ lies in the principal value range. If $x$ is outside $[-\pi/2, \pi/2]$, then $\sin^{-1}(\sin x)$ gives the angle in $[-\pi/2, \pi/2]$ that has the same sine as $x$ — not $x$ itself.

Understanding Fig. 2.2: The Graphs of $y = \cos x$ and $y = \cos^{-1} x$

The figure presents two panels side by side, showing how the cosine function and its inverse are related through reflection and domain restriction.

Panel (i): $y = \cos x$

The first panel shows the familiar cosine wave plotted on the $x$-$y$ plane. The horizontal axis ($X'OX$) carries $x$-marks at intervals of $\pi/2$, from $-5\pi/2$ to $5\pi/2$. Two dashed horizontal guide lines run at $y = 1$ and $y = -1$, marking the maximum and minimum values of cosine. The full curve oscillates between these bounds, crossing the $x$-axis at odd multiples of $\pi/2$.

What makes this figure special is the darkened, thicker arc drawn over the interval $[0, \pi]$. This arc runs from the point $(0, 1)$ down to $(\pi, -1)$. This is the principal-value branch of the cosine function — the restricted portion that makes the function one-to-one. On this restricted domain $[0, \pi]$, $\cos x$ takes every value in $[-1, 1]$ exactly once, so it becomes invertible.

Panel (ii): $y = \cos^{-1} x$

The second panel shows the inverse cosine function, obtained by reflecting the cosine curve across the line $y = x$. The domain is now $[-1, 1]$, indicated by dashed vertical lines at $x = -1$ and $x = 1$. The $y$-axis is marked at multiples of $\pi/2$.

The darkened segment of this curve runs from $(1, 0)$ rising to $(-1, \pi)$. This is the principal branch of $\cos^{-1} x$, with range $[0, \pi]$. Every point on this segment corresponds to a point on the darkened arc in panel (i), with coordinates swapped.

The Physical Idea

The figure teaches a fundamental concept about invertibility. Trigonometric functions are periodic and therefore not one-to-one over their natural domains. To define an inverse, we must restrict the domain to an interval where the function is strictly monotonic (here, decreasing from $1$ to $-1$). The inverse function then "undoes" the original: if $y = \cos^{-1} x$, then $\cos y = x$, provided $y$ lies in the principal range $[0, \pi]$.

The reflection across $y = x$ is not just a geometric trick — it embodies the algebraic relationship between a function and its inverse. Every point $(a, b)$ on the cosine curve becomes $(b, a)$ on the inverse cosine curve.

Key Formulas Developed from This Figure

The textbook uses this graphical understanding to establish the fundamental identities:

The first identity says: take any $x$ in $[-1, 1]$, find its angle $\cos^{-1} x$ in $[0, \pi]$, then take the cosine of that angle — you get back $x$. The second says: take any angle $y$ in $[0, \pi]$, compute $\cos y$, then find the inverse cosine of that value — you get back $y$.

How This Connects to Other Inverse Trigonometric Functions …

Understanding Fig. 2.3: The Cosecant and Its Inverse

The figure presents two carefully aligned panels that show the relationship between $y = \csc x$ and its inverse $y = \csc^{-1} x$. The left panel displays the cosecant function itself, while the right panel shows its inverse obtained by reflecting across the line $y = x$. This visual pairing is the standard method for understanding any inverse function: if $(a,b)$ lies on the graph of $y = \csc x$, then $(b,a)$ lies on the graph of $y = \csc^{-1} x$.

The Left Panel: $y = \csc x$

The cosecant function is defined as $\csc x = \frac{1}{\sin x}$, so it inherits the vertical asymptotes of the reciprocal wherever $\sin x = 0$. These occur at $x = n\pi$ for any integer $n$, and the figure shows dashed vertical lines at $x = -\pi, 0, \pi, 2\pi$ to mark these forbidden values. Between each pair of asymptotes, the graph forms a U-shaped branch that opens upward or downward. The branches touch the lines $y = 1$ and $y = -1$ at their minima and maxima respectively — these are the points where $\sin x = \pm 1$, so $\csc x = \pm 1$.

The axes are labelled with key points: the $x$-axis shows $-\pi, -\pi/2, \pi/2, \pi, 3\pi/2, 2\pi$, and the $y$-axis shows $2, 1, -1, -2$. The principal branch — the one chosen for defining the inverse — is darkened. This branch lies over the interval $[-\pi/2, \pi/2]$ with the point $x = 0$ removed. On this branch, for $x$ in $(0, \pi/2]$ we have $y \geq 1$, and for $x$ in $[-\pi/2, 0)$ we have $y \leq -1$.

The Right Panel: $y = \csc^{-1} x$

The inverse cosecant function is defined only for $|x| \geq 1$, since the range of $\csc x$ is $(-\infty, -1] \cup [1, \infty)$. The graph shows horizontal dashed asymptotes at $y = n\pi$, which are the reflections of the vertical asymptotes from the left panel. The two darkened arcs correspond to the principal value branch: one arc lies in $y \in (0, \pi/2]$ for $x \geq 1$, and the other lies in $y \in [-\pi/2, 0)$ for $x \leq -1$.

The Key Formula

The fundamental relationship that the figure illustrates is:

This means that finding $\csc^{-1} x$ asks: "What angle $y$ in the principal range has cosecant equal to $x$?" The figure makes this concrete by showing how each point on the darkened branch of $\csc x$ maps to a corresponding point on the darkened branch of $\csc^{-1} x$ through reflection across $y = x$.

Why This Matters for Problem Solving

When you encounter $\csc^{-1} x$ in an exam, the figure reminds you of three critical facts:

• The domain is $|x| \geq 1$ — never try to evaluate $\csc^{-1} x$ for values between $-1$ and $1$. …

Understanding Fig. 2.4: The secant and inverse secant pair

The figure presents two coordinated panels that show how the secant function and its inverse are related through reflection across the line $y = x$. This visual relationship is the same geometric principle that connects any function with its inverse — a concept introduced in Chapter 1 and now applied to trigonometric functions.

Panel (i): $y = \sec x$

The secant function, defined as $\sec x = \frac{1}{\cos x}$, has a domain that excludes all odd multiples of $\frac{\pi}{2}$ — these are the points where $\cos x = 0$. The graph shows vertical dashed asymptotes at $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$, and $x = \frac{3\pi}{2}$, marking where the function blows up to $\pm\infty$.

The curve consists of repeating U-shaped branches. Each branch opens upward (touching $y = 1$ at its minimum) or downward (touching $y = -1$ at its maximum), depending on the interval. The $x$-axis is marked at $-\pi$, $-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$; the $y$-axis shows $2$, $1$, $-1$, and $-2$.

The principal branch — the one we restrict to make the function one-to-one — is darkened. This branch lies over the interval $[0, \pi]$ with the point $\frac{\pi}{2}$ removed. On $[0, \frac{\pi}{2})$, the curve rises from $y = 1$ upward toward the asymptote; on $(\frac{\pi}{2}, \pi]$, it rises from $y = -1$ upward (becoming less negative) toward the asymptote. This restricted secant function is one-to-one and covers all values $|y| \geq 1$.

Panel (ii): $y = \sec^{-1} x$

The inverse secant function is obtained by reflecting the principal branch of $y = \sec x$ across the line $y = x$. Its domain is $|x| \geq 1$, matching the range of the restricted secant. The range of $\sec^{-1} x$ is $[0, \pi] - \{\frac{\pi}{2}\}$, which corresponds to the restricted domain of the secant.

The graph shows horizontal dashed asymptotes at $y = \frac{\pi}{2}$, $y = \frac{3\pi}{2}$, and $y = -\frac{\pi}{2}$, which are the reflections of the vertical asymptotes from the secant graph. The principal arcs are darkened: for $x \geq 1$, the curve lies in $[0, \frac{\pi}{2})$; for $x \leq -1$, it lies in $(\frac{\pi}{2}, \pi]$.

The central formula

The defining relationship between secant and its inverse is:

Here, $x$ is any real number with magnitude at least $1$, and $y$ is the angle (in radians) whose secant equals $x$, taken from the principal value branch. The restriction $y \in [0, \pi] - \{\frac{\pi}{2}\}$ is what makes the correspondence one-to-one — without it, infinitely many angles would give the same secant value.

What the figure teaches …

What Fig. 2.5 Shows

The figure is split into two panels placed side by side, showing how the tangent function and its inverse are related through reflection.

Panel (i): $y = \tan x$ — the familiar repeating curve with vertical dashed asymptotes at $x = -\frac{3\pi}{2}$, $-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\frac{3\pi}{2}$. The $x$-axis is marked at $-\pi$, $-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$; the $y$-axis at $-2$, $-1$, $1$, $2$. Each branch of the curve rises steeply from $-\infty$ to $+\infty$ between consecutive asymptotes. The central branch over $(-\pi/2, \pi/2)$, passing through the origin, is drawn in a darker line — this is the principal branch chosen for inversion.

Panel (ii): $y = \tan^{-1} x$ — the reflection of the principal branch across the line $y = x$. It is an increasing S-shaped curve defined for all real $x$, with horizontal dashed asymptotes at $y = \pm \pi/2$. The curve passes through the origin, and its range is exactly $(-\pi/2, \pi/2)$. The other branches of $\tan x$ (the lighter curves in panel i) reflect to corresponding branches of $\tan^{-1} x$ at $y = \pm 3\pi/2$, etc., but these are not part of the principal value.

The Core Idea

A function must be one-to-one to have an inverse. The tangent function on its full domain is not one-to-one — it repeats every $\pi$ and blows up at odd multiples of $\pi/2$. By restricting $\tan x$ to the open interval $(-\pi/2, \pi/2)$, we obtain a bijection onto $\mathbb{R}$. The inverse of this restricted function is $\tan^{-1} x$, also written $\arctan x$.

The textbook defines the principal value branch as:

This means: for any real number $x$, $\tan^{-1} x$ is the unique angle $y$ in $(-\pi/2, \pi/2)$ such that $\tan y = x$.

Key Relationships from the Figure

The reflection property gives two fundamental identities:

The second identity is the reason the principal branch is chosen: it ensures that applying $\tan^{-1}$ after $\tan$ returns the original angle, but only when that angle lies in the restricted interval. If $y$ is outside $(-\pi/2, \pi/2)$, $\tan^{-1}(\tan y)$ gives the angle in the principal branch that has the same tangent — not the original $y$.

What the Asymptotes Tell Us …

What Fig. 2.6 Shows

The figure has two panels, placed side by side, that together illustrate how the inverse cotangent function is constructed from the original cotangent function.

Panel (i): $y = \cot x$ — the familiar cotangent curve. The horizontal axis is $x$, the vertical axis is $y$. The graph consists of infinitely many repeating branches, each decreasing from left to right. Vertical dashed lines (asymptotes) appear at every integer multiple of $\pi$: $x = -\pi, 0, \pi, 2\pi$. The curve crosses the $x$-axis at $x = -\pi/2,\ \pi/2,\ 3\pi/2$. The axes are marked at $x = -\pi/2,\ \pi/2,\ \pi,\ 3\pi/2,\ 2\pi$ and at $y = -2,\ -1,\ 1,\ 2$. One particular branch — the one lying over the open interval $(0,\pi)$ — is drawn more darkly than the others. This darkened branch passes through the point $(\pi/2,\ 0)$ and is called the principal branch of $\cot x$.

Panel (ii): $y = \cot^{-1} x$ — the inverse cotangent function. This graph is obtained by reflecting the principal branch of $\cot x$ across the line $y = x$. The axes are now swapped: the horizontal axis is $x$, the vertical axis is $y$. The curve is defined for all real $x$ (the entire $x$-axis) and is strictly decreasing. Horizontal dashed lines (asymptotes) appear at $y = 0,\ \pi,\ 2\pi$. The darkened central branch — the one that corresponds to the reflection of the principal branch from panel (i) — runs between $y = 0$ and $y = \pi$, passing through the point $(0,\ \pi/2)$. This darkened branch is the principal value branch of $\cot^{-1} x$.

The Physical Idea

The figure teaches a fundamental concept about inverse functions: to define an inverse for a periodic function like $\cot x$, we must first restrict the original function to an interval where it is one-to-one (bijective). The cotangent function on its full domain is not one-to-one — it repeats every $\pi$ units. But if we restrict it to the open interval $(0,\ \pi)$, it becomes strictly decreasing and covers every real number exactly once. This restricted function has an inverse, and that inverse is $\cot^{-1} x$.

The reflection across $y = x$ is the geometric way of swapping the roles of input and output. Every point $(a,\ b)$ on the principal branch of $\cot x$ becomes the point $(b,\ a)$ on the principal value branch of $\cot^{-1} x$. For example, $(\pi/2,\ 0)$ on $\cot x$ becomes $(0,\ \pi/2)$ on $\cot^{-1} x$.

Key Formulas

The textbook establishes the following central relationships from this figure:

This notation means: the inverse cotangent function takes any real number as input and returns an angle in the open interval $(0,\ \pi)$ as output. The domain is all real numbers; the range (principal value branch) is $(0,\ \pi)$.

The defining relation is:

In words: if $y$ is the inverse cotangent of $x$, then $y$ is the angle in $(0,\ \pi)$ whose cotangent equals $x$.

Two important composition identities follow directly:

The first says: take any real $x$, find its inverse cotangent (an angle in $(0,\ \pi)$), then take the cotangent of that angle — you get back $x$. The second says: start with any angle $y$ in $(0,\ \pi)$, take its cotangent, then take the inverse cotangent of that result — you get back $y$. …