Trigonometric Functions, Inverse Trigonometry, and Introduction to Limits

Course Logistics and Exam Guidelines

• Class Schedule Details:

  • Friday class periods are short (single period duration).

  • The upcoming exam takes place on Wednesday during the full class period.

• Homework and Notes Policy:

  • There is no daily graded homework; assessment preparation relies on quizzes.

  • Online versions of lecture notes are not provided. The course textbook serves as the primary reference resource.

  • Note-taking strategy recommendation: Model note-taking directly after the board work standard, as structured written mathematics reflects refined communication practices.

• Exam Mechanics and Calculator Policy:

  • Calculators are strictly prohibited on exams to avoid security issues and administrative complexities.

  • Exam questions are explicitly designed to be completed without numerical computational tools.

  • Simplification standards:

    • Answers do not need to be simplified or converted into decimal or colon forms.

    • Exact trigonometric or functional values must be left as exact expressions (e.g., an expression like $\sin(4)$ is a valid final exact number).

• Exam Format and Preparation:

  • The exam consists of exactly $4$ questions.

  • Students are given the entire class period to complete the exam.

  • Question structure: Exam problems are structured identically to quiz problems, which in turn reflect in-class examples.

  • Mastery of all quiz problems is sufficient preparation for the exam.

  • Required materials: Writing utensil and critical reasoning.

Right Triangle Definitions of Trigonometric Functions

• Right Triangle Construction:

  • A right triangle contains one $90^\text{deg}$ angle and two acute angles.

  • Let $\theta$ represent one of the acute angles.

  • Let $x$ represent the horizontal leg length.

  • Let $y$ represent the vertical leg length.

  • Let $r$ represent the hypotenuse length.

• Pythagorean Relationship:

  • The relationship between the side lengths is governed by the Pythagorean Theorem:         $x^2 + y^2 = r^2$

• Primary Trigonometric Functions:

  • Sine Function:         $\sin(\theta) = \frac{y}{r} = \frac{\text{opposite}}{\text{hypotenuse}}$

  • Cosine Function:         $\cos(\theta) = \frac{x}{r} = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • Tangent Function:         $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} = \frac{\text{opposite}}{\text{adjacent}}$

Circle-Based Definitions and the Right-Hand Rule

• Geometric Setup on a Circle:

  • Consider a circle centered at the origin $(0, 0)$ with radius $r$.

  • Select a point $(x, y)$ on the circle at distance $r$ from the origin.

  • Define the angle $\theta$ between the positive x-axis and the radial line segment connecting the origin to $(x, y)$.

• Rules for Standard Angle Measurement:

  • Initial side: Must always start on the positive x-axis (never the negative x-axis).

  • Direction: Positive angle measurements move in the counterclockwise (CCW) direction.

  • Right-Hand Rule:

    • Using the right hand, curl the fingers in the direction of counterclockwise rotation (positive angular displacement).

    • The thumb points outwards, perpendicular to the plane (towards the observer).

    • Using the left hand gives the opposite direction (pointing away), illustrating why the right-hand convention is standard.

• Parametric Representation of Circle Coordinates:

  • The coordinates of the point on the circle are:         $(x, y) = (r \times \cos(\theta), r \times \sin(\theta))$

  • For a unit circle where $r = 1$, the coordinates simplify to:         $(x, y) = (\cos(\theta), \sin(\theta))$

• Equivalence to Triangle Definitions:

  • Constructing a vertical segment from $(x, y)$ down to the x-axis forms a right triangle with base $x = r \times \cos(\theta)$, height $y = r \times \sin(\theta)$, and hypotenuse $r$

  • Rearranging these coordinate equations yields the classical ratios:         $\cos(\theta) = \frac{x}{r}$         $\sin(\theta) = \frac{y}{r}$

• Generality of the Circle Definition:

  • The circle definition is superior because it applies to arbitrary angles beyond acute right-triangle constraints ($\theta > 90^\circ$).

  • It defines trigonometric functions for:

    • $\theta = 180^\circ$

    • $\theta = 270^\circ$

    • Negative rotations (e.g., $-45^\circ$

    • Rotations exceeding $360^\circ$

Inverse Trigonometric Functions and Restricted Domains

• Domain Restrictions for Invertibility:

  • Full trigonometric functions repeat outputs around the circle, failing the horizontal line test.

  • To construct single-valued inverse functions, domains are restricted to continuous intervals yielding unique outputs:

    • Cosine domain restriction: $\theta \in [0, \pi]$

    • Sine domain restriction: $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

• Conceptual Meaning of Inverse Functions:

  • Notation: The superscript $-1$ in $\cos^{-1}(x)$ or $\sin^{-1}(x)$ designates a functional inverse, not an exponent or reciprocal.         $\cos^{-1}(x) \neq \frac{1}{\cos(x)}$

  • Alternative notation: $\arccos(x)$ or arccos represents inverse cosine.

  • Conceptual core: An inverse trigonometric expression represents an angle within the restricted domain whose trigonometric output equals the input ratio.

• Key Composition Properties:

  • For a general function $f$ and its inverse $f^{-1}$:         $f(f^{-1}(x)) = x$         $f^{-1}(f(x)) = x$

  • For the cosine function on its domain:         $\cos(\cos^{-1}(x)) = x$

Geometric Visualization and Composite Inverse Trig Evaluations

• Methodology for Composite Expressions:

  • To evaluate an expression like $\tan(\cos^{-1}(a))$, treat the inner inverse function as an angle: $\theta = \cos^{-1}(a)$.

  • Translate the algebraic relation into a geometric right triangle.

  • Use the Pythagorean theorem to derive the missing side length.

  • Evaluate the outer function directly from the triangle ratios.

  • Pedagogical note on visualization: Drawing physical geometric representations helps unpack abstract mathematical definitions from the inside out.

• Example 1: Evaluating $\tan\left(\cos^{-1}\left(\frac{3}{7}\right)\right)$

  • Define the angle: Let $\theta = \cos^{-1}\left(\frac{3}{7}\right)$, which implies $\cos(\theta) = \frac{3}{7}$.

  • Construct the right triangle:

    • Adjacent leg $x = 3$

    • Hypotenuse $r = 7$

  • Determine vertical leg $y$ via Pythagorean Theorem:         $3^2 + y^2 = 7^2$         $y^2 = 7^2 - 3^2 = 49 - 9 = 40$         $y = \sqrt{40}$         (Take the positive principal square root because geometric leg lengths are positive distances).

  • Evaluate functions of $\theta$:         $\sin(\theta) = \frac{\sqrt{40}}{7}$         $\tan(\theta) = \frac{\sqrt{40}}{3}$

  • Simplification of radical expressions:         $\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$         $\tan\left(\cos^{-1}\left(\frac{3}{7}\right)\right) = \frac{2\sqrt{10}}{3}$

  • Both $\frac{\sqrt{40}}{3}$ and $\frac{2\sqrt{10}}{3}$ represent valid equal quantities.

• Example 2: Evaluating $\cos\left(\sin^{-1}\left(\frac{2}{9}\right)\right)$

  • Define the angle: Let $\theta = \sin^{-1}\left(\frac{2}{9}\right)$, which implies $\sin(\theta) = \frac{2}{9}$.

  • Construct the right triangle:

    • Opposite leg $y = 2$

    • Hypotenuse $r = 9$

  • Determine adjacent leg $x$:         $x^2 + 2^2 = 9^2$         $x^2 = 9^2 - 2^2 = 81 - 4 = 77$         $x = \sqrt{77}$

  • Since $77 = 11 \times 7$, there are no perfect square factors to pull out of the radical.

  • Evaluate cosine of the angle:         $\cos\left(\sin^{-1}\left(\frac{2}{9}\right)\right) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{77}}{9}$

Introduction to Limits and Algebraic Resolution

• Conceptual Definition of a Limit:

  • A limit formalizes the behavior of a function $f(x)$ as the input $x$ approaches a target value $a$.

  • Standard notation:         $\lim_{x \to a} f(x)$

  • Reads: "The limit as $x$ approaches $a$ of $f(x)$."

  • A limit can exist and equal a concrete number $L$ even if the function itself is undefined at $x = a$ ($f(a)$ does not exist).

• Analyzing Discontinuities and Rational Functions:

  • Consider the rational function:         $f(x) = \frac{x^2 - 6x + 8}{x^2 - 4}$

  • Domain determination:

    • Setting denominator to zero: $x^2 - 4 = 0 \implies x = \pm 2$

    • The function $f(x)$ is undefined at $x = 2$ and $x = -2$ due to division by zero.

  • Evaluating Function Behavior near Undefined Points:

    • Standard point plotting ("plug and chug") works everywhere on the domain except at domain boundaries/holes ($x = \pm 2$).

    • Limits determine whether an undefined point corresponds to a vertical asymptote or a removable hole.

• Evaluating $\lim_{x \to 2} \frac{x^2 - 6x + 8}{x^2 - 4}$ via Factoring:

  • Step 1: Factor the numerator:         $x^2 - 6x + 8 = (x - 4)(x - 2)$

  • Step 2: Factor the denominator:         $x^2 - 4 = (x + 2)(x - 2)$

  • Step 3: Simplify the expression for all $x \neq 2$:         $f(x) = \frac{(x - 4)(x - 2)}{(x + 2)(x - 2)} = \frac{x - 4}{x + 2} \quad \text{for } x \neq 2$

  • Step 4: Compute the limit by direct substitution into the reduced expression:         $\lim_{x \to 2} f(x) = \lim_{x \to 2} \frac{x - 4}{x + 2} = \frac{2 - 4}{2 + 2} = \frac{-2}{4} = -\frac{1}{2}$

  • Conclusion:

    • $f(2)$ is undefined.

    • The limit as $x \to 2$ is $-\frac{1}{2}$.

    • Geometrically, the graph of $f(x)$ possesses a removable hole at the point $\left(2, -\frac{1}{2}\right)$.