4.2 Graphs of Trigonometric Functions & Inverse Trigonometric Functions

Graphs of Trigonometric Functions & Inverse Trigonometric Functions

In the FSc Part I Mathematics curriculum and the UET ECAT syllabus, the study of periodic waveforms, graph topologies, and inverse trigonometric mappings occupies a central position. In engineering disciplines—such as electrical circuit analysis, structural vibration modeling, and mechanical kinematics—waveforms are modeled by periodic circular functions, while inverse functions are required to extract phase angles and orientations. ECAT questions frequently challenge candidates to determine the fundamental periods of composite expressions, evaluate principal values outside standard branches, and simplify algebraic inverse summations without calculator assistance.

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1. Periodic Functions and Fundamental Periods

A function $f(x)$ is termed periodic if there exists a positive real number $p > 0$ such that:

$f(x + p) = f(x)$

for all $x$ in the domain of $f$. The smallest positive number $p$ satisfying this condition is called the primitive period or fundamental period of the function.

Fundamental Periods of the Basic Functions

1. Sine and Cosine: From circular rotations, adding a complete revolution of $2\pi$ radians returns a point $P(x, y)$ to its identical position on the unit circle: $\sin(x + 2\pi) = \sin x, \quad \cos(x + 2\pi) = \cos x$ No positive value smaller than $2\pi$ satisfies this identity for all $x$. Thus, the fundamental period of $\sin x$ and $\cos x$ is $2\pi$.
2. Reciprocal Functions: Because $\csc x = \frac{1}{\sin x}$ and $\sec x = \frac{1}{\cos x}$, their periods are identically $2\pi$.
3. Tangent and Cotangent: In the Cartesian plane, tangent represents the slope of the line passing through the origin and $P(x, y)$: $\tan(x + \pi) = \frac{\sin(x + \pi)}{\cos(x + \pi)} = \frac{-\sin x}{-\cos x} = \frac{\sin x}{\cos x} = \tan x$ The slope repeats every half-revolution ($\pi$ radians). Thus, the fundamental period of $\tan x$ and $\cot x$ is $\pi$.

The Argument Scaling Theorem

If a function $f(x)$ has a fundamental period $T$, then for any non-zero real scalar $k$:

$\text{Period of } f(kx) = \frac{T}{|k|}$

This horizontal compression/dilation theorem leads to immediate ECAT solutions:

• $\sin\left(\frac{x}{4}\right) \implies T' = \frac{2\pi}{1/4} = 8\pi$
• $\tan(3x) \implies T' = \frac{\pi}{3}$
• $\cos\left(\frac{2x}{7}\right) \implies T' = \frac{2\pi}{2/7} = 7\pi$

Periodic Behavior of Powers and Absolute Values

When trigonometric functions are squared or placed in absolute values, negative lobes reflect across the x-axis, effectively halving their periods:

• $\sin^2(kx) = \frac{1 - \cos(2kx)}{2} \implies \text{Period} = \frac{2\pi}{2|k|} = \frac{\pi}{|k|}$
• $|\sin(kx)|$ and $|\cos(kx)| \implies \text{Period} = \frac{\pi}{|k|}$
• $|\tan(kx)| \implies \text{Period} = \frac{\pi}{|k|}$ (tangent period is already $\pi$)

Period of Linear Combinations (The LCM Rule)

If $f(x) = g(x) \pm h(x)$, where $g(x)$ has rational period $T_1 = \frac{a}{b}\pi$ and $h(x)$ has rational period $T_2 = \frac{c}{d}\pi$, the period of the combination is the least common multiple:

$\text{Period} = \text{LCM}(T_1, T_2) = \pi \times \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)}$

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2. Graph Topologies, Domains, Ranges, and Asymptotes

Understanding the continuous and discontinuous features of trigonometric graphs enables rapid visualization of function behavior on the ECAT.

Comprehensive Comparison of Trigonometric Functions

Key Geometric Features

• Roots (x-intercepts):
  • $\sin x = 0 \iff x = n\pi$
  • $\cos x = 0 \iff x = (2n+1)\frac{\pi}{2}$
  • $\tan x = 0 \iff x = n\pi$
  • $\cot x = 0 \iff x = (2n+1)\frac{\pi}{2}$
• Discontinuities & Asymptotes:
  • $\tan x$ and $\sec x$ diverge to $\pm \infty$ at odd multiples of $\frac{\pi}{2}$ due to division by $\cos x = 0$.
  • $\cot x$ and $\csc x$ diverge to $\pm \infty$ at integer multiples of $\pi$ due to division by $\sin x = 0$.

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3. The Restricted Domain Requirement for Inverse Functions

A function $f$ possesses an inverse $f^{-1}$ if and only if it is bijective (one-to-one and onto). Because trigonometric functions are periodic, every output value is repeated infinitely many times across $(-\infty, \infty)$, completely failing the Horizontal Line Test.

To construct mathematical inverse functions, we restrict the domain of each circular function to a continuous interval on which it is strictly monotonic (strictly increasing or strictly decreasing) while retaining its complete range. The resulting inverse functions are termed the Principal Inverse Trigonometric Functions.

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4. The Principal Inverse Trigonometric Functions

1. Principal Inverse Sine ($\operatorname{Arcsin} x$ or $\sin^{-1} x$)

• Restricted Base Domain: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
• Inverse Domain: $[-1, 1]$
• Principal Range: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
• Quadrant Mapping: Maps positive ratios $x \in [0, 1]$ to Quadrant I $\left[0, \frac{\pi}{2}\right]$; maps negative ratios $x \in [-1, 0)$ to Quadrant IV $\left[-\frac{\pi}{2}, 0\right)$.

2. Principal Inverse Cosine ($\operatorname{Arccos} x$ or $\cos^{-1} x$)

• Restricted Base Domain: $[0, \pi]$
• Inverse Domain: $[-1, 1]$
• Principal Range: $[0, \pi]$
• Quadrant Mapping: Maps positive ratios $x \in [0, 1]$ to Quadrant I $\left[0, \frac{\pi}{2}\right]$; maps negative ratios $x \in [-1, 0)$ to Quadrant II $\left(\frac{\pi}{2}, \pi\right]$.

3. Principal Inverse Tangent ($\operatorname{Arctan} x$ or $\tan^{-1} x$)

• Restricted Base Domain: $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
• Inverse Domain: $\mathbb{R} = (-\infty, \infty)$
• Principal Range: $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
• Horizontal Asymptotes: $y = \frac{\pi}{2}$ as $x \to \infty$, and $y = -\frac{\pi}{2}$ as $x \to -\infty$.

4. Principal Inverse Cotangent, Secant, and Cosecant

Under the standard conventions used in Intermediate textbooks:

• $\operatorname{Arccot} x$: Domain $(-\infty, \infty)$, Range $(0, \pi)$.
• $\operatorname{Arcsec} x$: Domain $(-\infty, -1] \cup [1, \infty)$, Range $[0, \pi] \setminus \left\{\frac{\pi}{2}\right\}$.
• $\operatorname{Arccsc} x$: Domain $(-\infty, -1] \cup [1, \infty)$, Range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \setminus \{0\}$.

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5. Reflection Symmetries & Complementary Identities

Parity and Negative Input Symmetries

A critical distinction on the ECAT is how inverse functions handle negative arguments. Because the ranges of $\arcsin x$ and $\arctan x$ are symmetric around the origin ($[-\frac{\pi}{2}, \frac{\pi}{2}]$), they are odd functions:

$\arcsin(-x) = -\arcsin x$

$\arctan(-x) = -\arctan x$

$\operatorname{arccsc}(-x) = -\operatorname{arccsc} x$

In contrast, the principal range of $\arccos x$ is $[0, \pi]$, which is strictly non-negative. For negative inputs, $\arccos(-x)$ must produce an angle in Quadrant II:

$\arccos(-x) = \pi - \arccos x$

$\operatorname{arccot}(-x) = \pi - \operatorname{arccot} x$

$\operatorname{arcsec}(-x) = \pi - \operatorname{arcsec} x$

ECAT Example: Evaluate $\arccos\left(-\frac{1}{2}\right)$. Rather than guessing signs, write $\arccos\left(-\frac{1}{2}\right) = \pi - \arccos\left(\frac{1}{2}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$.

Complementary Identities

For every $x$ in the stated domain:

$\arcsin x + \arccos x = \frac{\pi}{2} \quad (-1 \le x \le 1)$

$\arctan x + \operatorname{arccot} x = \frac{\pi}{2} \quad (x \in \mathbb{R})$

$\operatorname{arcsec} x + \operatorname{arccsc} x = \frac{\pi}{2} \quad (|x| \ge 1)$

Why it works: if $\theta = \arcsin x$, then $x = \sin\theta = \cos\left(\frac{\pi}{2} - \theta\right)$, and $\frac{\pi}{2} - \theta$ lies in $[0, \pi]$, the principal range of $\arccos$. So $\arccos x = \frac{\pi}{2} - \theta$.

ECAT Example: If $\arcsin x = \frac{\pi}{5}$, then $\arccos x = \frac{\pi}{2} - \frac{\pi}{5} = \frac{3\pi}{10}$. There is no need to find $x$.

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6. Composite Expressions and Principal Values

$\arcsin(\sin\theta) = \theta$ only when $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, $\arccos(\cos\theta) = \theta$ only when $\theta \in [0, \pi]$, and $\arctan(\tan\theta) = \theta$ only when $\theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Outside those ranges, first rewrite the inner function with an angle that lies in the principal range.

The reverse composition is simpler: $\sin(\arcsin x) = x$ for every $x \in [-1, 1]$, and $\tan(\arctan x) = x$ for every real $x$.

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7. Addition Theorems for Inverse Tangent

For real $x$ and $y$:

$\arctan x + \arctan y = \begin{cases} \arctan\left(\dfrac{x+y}{1-xy}\right), & xy < 1 \\ \pi + \arctan\left(\dfrac{x+y}{1-xy}\right), & xy > 1,\ x > 0 \\ -\pi + \arctan\left(\dfrac{x+y}{1-xy}\right), & xy > 1,\ x < 0 \end{cases}$

$\arctan x - \arctan y = \arctan\left(\frac{x-y}{1+xy}\right) \quad (xy > -1)$

$2\arctan x = \arctan\left(\frac{2x}{1-x^2}\right) \quad (|x| < 1)$

ECAT Trap Example: Evaluate $\arctan 2 + \arctan 3$. Here $xy = 6 > 1$ and $x > 0$, so the correction term is needed: $\arctan 2 + \arctan 3 = \pi + \arctan\left(\frac{5}{1-6}\right) = \pi + \arctan(-1) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$ Forgetting the $\pi$ gives $-\frac{\pi}{4}$, which is impossible because both angles are positive.

A matching result for inverse sine, used less often, is $\arcsin x + \arcsin y = \arcsin\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right)$ when $x, y \ge 0$ and $x^2 + y^2 \le 1$.