Trigonometry Chapter 4 Comprehensive Study Guide on Trigonometric Functions, Graphs, Transformations, and Inverses

Basic Trigonometric Graphs and Properties

• The Sine Graph ($y = \sin(x)$):

  • Construction and Ordered Pairs: The graph of $y = \sin(x)$ is formed by plotting ordered pairs $(x, y)$ satisfying the equation and connecting them with a smooth curve.

  • Table of Fundamental Values:

  • When $x = 0$, $y = \sin(0) = 0$

  • When $x = \frac{\pi}{4}$, $y = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.707$

  • When $x = \frac{\pi}{2}$, $y = \sin\left(\frac{\pi}{2}\right) = 1$

  • When $x = \frac{3\pi}{4}$, $y = \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.707$

  • When $x = \pi$, $y = \sin(\pi) = 0$

  • When $x = \frac{5\pi}{4}$, $y = \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \approx -0.707$

  • When $x = \frac{3\pi}{2}$, $y = \sin\left(\frac{3\pi}{2}\right) = -1$

  • When $x = \frac{7\pi}{4}$, $y = \sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2} \approx -0.707$

  • When $x = 2\pi$, $y = \sin(2\pi) = 0$

  • Unit Circle Definition (Definition III):

  • Let $(x, y)$ be a point on the unit circle at a distance of $t$ units from the starting point $(1, 0)$ along the circumference. Then, $\sin(t) = y$.

  • Starting at $(1, 0)$ and traveling once around the unit circle (a total distance of $2\pi$ units), the value of $y$ in $y = \sin(t)$ corresponds directly to the y-coordinates of points $t$ units from $(1, 0)$.

  • Quadrant-by-Quadrant Trajectory:

  • Quadrant I: As $t$ increases from $0$ to $\frac{\pi}{2}$, point $P$ travels from $(1, 0)$ to $(0, 1)$, and $y = \sin(t)$ increases from $0$ to $1$

  • Quadrant II: As $t$ increases from $\frac{\pi}{2}$ to $\pi$, point $P$ travels in Quadrant II, and $y = \sin(t)$ decreases from $1$ back to $0$

  • Quadrant III: As $t$ increases from $\pi$ to $\frac{3\pi}{2}$, the length of vertical segment $AP$ increases from $0$ to $1$. Because it lies below the x-axis, the y-coordinate is negative, so $y = \sin(t)$ decreases from $0$ to $-1$

  • Quadrant IV: As $t$ increases from $\frac{3\pi}{2}$ to $2\pi$, point $P$ returns to $(1, 0)$, and $y = \sin(t)$ increases from $-1$ back to $0$

  • Key Properties of the Sine Function:

  • Range Bounds: The graph never goes above $1$ or below $-1$. The range is $[-1, 1]$, meaning $-1 \le \sin(x) \le 1$

  • Domain: The domain consists of all real numbers, or $(-\infty, \infty)$

  • Periodicity: The graph repeats itself every $2\pi$ units along the x-axis. The period is $p = 2\pi$, which is the smallest positive number such that $\sin(x + p) = \sin(x)$ for all $x$

  • Amplitude: The amplitude is $1$

  • Zeros: The function has an infinite number of zeros (x-intercepts) located at $x = k\pi$ for any integer $k$

• The Cosine Graph ($y = \cos(x)$):

  • Unit Circle Derivation: By Definition III, if $(x, y)$ is $t$ units from $(1, 0)$ along the circumference of the unit circle, then $\cos(t) = x$.

  • Visualization via Circle Rotation: To visualize how x-coordinates generate the cosine graph, rotating the unit circle $90^\circ$ counterclockwise allows x-coordinates to be represented as vertical line segments.

  • Table of Fundamental Values:

  • When $x = 0$, $y = \cos(0) = 1$

  • When $x = \frac{\pi}{4}$, $y = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

  • When $x = \frac{\pi}{2}$, $y = \cos\left(\frac{\pi}{2}\right) = 0$

  • When $x = \frac{3\pi}{4}$, $y = \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

  • When $x = \pi$, $y = \cos(\pi) = -1$

  • When $x = \frac{5\pi}{4}$, $y = \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

  • When $x = \frac{3\pi}{2}$, $y = \cos\left(\frac{3\pi}{2}\right) = 0$

  • When $x = \frac{7\pi}{4}$, $y = \cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}$

  • When $x = 2\pi$, $y = \cos(2\pi) = 1$

  • Key Properties of the Cosine Function:

  • Domain: All real numbers

  • Range: $[-1, 1]$

  • Amplitude: $1$

  • Period: $2\pi$

  • Zeros: The zeros (x-intercepts) occur at $x = \frac{\pi}{2} + k\pi$ for any integer $k$

• The Tangent Graph ($y = \tan(x)$):

  • Definition and Undefined Points: Because $\tan(x) = \frac{\sin(x)}{\cos(x)}$, the tangent function is undefined whenever $\cos(x) = 0$. Specifically, it is undefined at $x = \frac{\pi}{2}$ due to division by zero.

  • Vertical Asymptotes: Dotted vertical lines called asymptotes are located at $x = \frac{\pi}{2} + k\pi$ for any integer $k$. The graph never crosses or touches these lines.

  • Asymptotic Behavior:

  • For values of $x$ immediately to the left of $\frac{\pi}{2}$, $\tan(x)$ becomes extremely large in the positive direction.

  • For values of $x$ immediately to the right of $\frac{\pi}{2}$, $\tan(x)$ becomes extremely large in the negative direction.

  • Key Properties of the Tangent Function:

  • Domain: All real $x \neq \frac{\pi}{2} + k\pi$ for any integer $k$

  • Range: All real numbers, $(-\infty, \infty)$

  • Amplitude: Not defined (there is no highest or lowest point on the graph)

  • Period: $\pi$

  • Zeros: $x = k\pi$ for any integer $k$ (identical to the zeros of $\sin(x)$)

  • Asymptotes: $x = \frac{\pi}{2} + k\pi$ for any integer $k$

• The Cosecant Graph ($y = \csc(x)$):

  • Reciprocal Relationship: $y = \csc(x) = \frac{1}{\sin(x)}$. The function is undefined whenever $\sin(x) = 0$.

  • Table of Fundamental Values:

  • When $x = 0$, $\sin(0) = 0$, so $\csc(0)$ is undefined

  • When $x = \frac{\pi}{4}$, $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, so $\csc\left(\frac{\pi}{4}\right) = \sqrt{2} \approx 1.4$

  • When $x = \frac{\pi}{2}$, $\sin\left(\frac{\pi}{2}\right) = 1$, so $\csc\left(\frac{\pi}{2}\right) = 1$

  • When $x = \frac{3\pi}{4}$, $\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}$, so $\csc\left(\frac{3\pi}{4}\right) = \sqrt{2} \approx 1.4$

  • When $x = \pi$, $\sin(\pi) = 0$, so $\csc(\pi)$ is undefined

  • When $x = \frac{5\pi}{4}$, $\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$, so $\csc\left(\frac{5\pi}{4}\right) = -\sqrt{2} \approx -1.4$

  • When $x = \frac{3\pi}{2}$, $\sin\left(\frac{3\pi}{2}\right) = -1$, so $\csc\left(\frac{3\pi}{2}\right) = -1$

  • When $x = \frac{7\pi}{4}$, $\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}$, so $\csc\left(\frac{7\pi}{4}\right) = -\sqrt{2} \approx -1.4$

  • When $x = 2\pi$, $\sin(2\pi) = 0$, so $\csc(2\pi)$ is undefined

  • Key Properties of the Cosecant Function:

  • Domain: All real $x \neq k\pi$ for any integer $k$

  • Range: $y \le -1$ or $y \ge 1$, or in interval notation $(-\infty, -1] \cup [1, \infty)$

  • Amplitude: Not defined

  • Period: $2\pi$

  • Zeros: None (the graph never crosses the x-axis)

  • Asymptotes: $x = k\pi$ for any integer $k$

• The Cotangent Graph ($y = \cot(x)$):

  • Reciprocal Relationship: $y = \cot(x) = \frac{\cos(x)}{\sin(x)}$.

  • Key Properties:

  • Domain: All real $x \neq k\pi$ for any integer $k$

  • Range: All real numbers

  • Amplitude: Not defined

  • Period: $\pi$

  • Zeros: $x = \frac{\pi}{2} + k\pi$ for any integer $k$

  • Asymptotes: $x = k\pi$ for any integer $k$

• The Secant Graph ($y = \sec(x)$):

  • Reciprocal Relationship: $y = \sec(x) = \frac{1}{\cos(x)}$.

  • Key Properties:

  • Domain: All real $x \neq \frac{\pi}{2} + k\pi$ for any integer $k$

  • Range: $y \le -1$ or $y \ge 1$

  • Amplitude: Not defined

  • Period: $2\pi$

  • Zeros: None

  • Asymptotes: $x = \frac{\pi}{2} + k\pi$ for any integer $k$

Symmetry and Even/Odd Trigonometric Relationships

• Definitions of Symmetry:

  • Even Function: A function for which replacing $x$ with $-x$ leaves the defining expression unchanged, such that $f(-x) = f(x)$. If a point $(x, y)$ is on the graph, the point $(-x, y)$ is also on the graph. The graph is symmetric across the y-axis.

  • Odd Function: A function for which replacing $x$ with $-x$ changes the sign of the defining expression, such that $f(-x) = -f(x)$. If a point $(x, y)$ is on the graph, the point $(-x, -y)$ is also on the graph. The graph is symmetric about the origin.

• Unit Circle Geometric Derivation:

  • Consider an angle $\theta$ and its opposite $\theta' = -\theta$ drawn in standard position on the unit circle.

  • Let the terminal side of $\theta$ intersect the unit circle at $(x, y)$ and the terminal side of $-\theta$ intersect at $(x, -y)$.

  • On the unit circle, $\cos(\theta) = x$ and $\sin(\theta) = y$

  • Evaluating $\theta' = -\theta$:

  • $\cos(-\theta) = x = \cos(\theta)$, demonstrating that cosine is an even function

  • $\sin(-\theta) = -y = -\sin(\theta)$, demonstrating that sine is an odd function

• Summary of All Six Trigonometric Functions:

  • Sine: Odd function, $\sin(-\theta) = -\sin(\theta)$

  • Cosine: Even function, $\cos(-\theta) = \cos(\theta)$

  • Tangent: Odd function, $\tan(-\theta) = -\tan(\theta)$

  • Cosecant: Odd function, $\csc(-\theta) = -\csc(\theta)$

  • Secant: Even function, $\sec(-\theta) = \sec(\theta)$

  • Cotangent: Odd function, $\cot(-\theta) = -\cot(\theta)$

Amplitude, Reflection, and Period Transformations

• Amplitude Stretch:

  • For equations of the form $y = A \sin(x)$ or $y = A \cos(x)$, the coefficient $A$ acts as a vertical stretch factor.

  • The amplitude is defined as $|A|$.

  • Example: For $y = 2 \sin(x)$, the amplitude is $2$, stretching the maximum values to $2$ and minimum values to $-2$ over $0 \le x \le 2\pi$.

• Reflection About the x-Axis:

  • If $A < 0$, the graphs of $y = A \sin(x)$ and $y = A \cos(x)$ are reflected across the x-axis.

  • The amplitude remains positive and equals $|A|$.

  • Example: For $y = -2 \cos(x)$, the graph of $y = 2 \cos(x)$ is inverted (reflected across the x-axis) over the domain $-2\pi \le x \le 4\pi$. Peak values at $y = 2$ become troughs at $y = -2$.

• Period Transformations:

  • Argument Definition: The input variable or expression inside the trigonometric function is formally called the argument.

  • Cycle Requirement: For functions $y = \sin(Bx)$ or $y = \cos(Bx)$ to complete one single basic cycle, the argument $Bx$ must vary from $0$ to $2\pi$

  • $0 \le Bx \le 2\pi \implies 0 \le x \le \frac{2\pi}{B}$

  • Period Formula: Assuming $B > 0$, the period is $\text{Period} = \frac{2\pi}{B}$.

  • Frequency: The graph completes $B$ full cycles within a distance of $2\pi$ units on the x-axis.

  • Negative Coefficient $B$ ($B < 0$): Use the properties of even and odd functions to rewrite the function so that $B$ becomes positive prior to graphing.

  • Example: For $y = \sin(2x)$ over $0 \le x \le 2\pi$, $B = 2$. The period is $\frac{2\pi}{2} = \pi$, so the graph completes $2$ full cycles within $2\pi$ units.

Vertical and Horizontal Translations and Phase Shift

• Vertical Translations:

  • The graph of $y = f(x) + k$ represents the graph of $y = f(x)$ translated $k$ units vertically.

  • If $k > 0$, the graph shifts up by $k$ units.

  • If $k < 0$, the graph shifts down by $|k|$ units.

  • Real-World Application (Ferris Wheel Model):

  • A Ferris wheel has a diameter of $250\,\text{ft}$, a ground clearance at the bottom of $14\,\text{ft}$, and completes one revolution every $20\,\text{minutes}$.

  • The midline height is $\text{radius} + \text{clearance} = 125 + 14 = 139\,\text{ft}$.

  • The height of a rider is modeled using a vertically shifted trigonometric function where $k = 139$.

• Horizontal Translations (Phase Shift):

  • Adding a term to the argument of a function yields a horizontal translation.

  • General Form: $y = A \sin(Bx + C)$ or $y = A \cos(Bx + C)$.

  • Determining One Basic Cycle: Set the argument $Bx + C$ to vary between $0$ and $2\pi$:

  • $0 \le Bx + C \le 2\pi$

  • $-C \le Bx \le 2\pi - C$

  • $-\frac{C}{B} \le x \le \frac{2\pi - C}{B}$ (assuming $B > 0$)

  • Horizontal Shift Formula: The value of $x$ at which the basic cycle begins is $-\frac{C}{B}$. This is the horizontal shift (or phase shift).

  • If $-\frac{C}{B} < 0$, the graph shifts to the left.

  • If $-\frac{C}{B} > 0$, the graph shifts to the right.

  • Period Verification: The period is the length of the cycle interval: $\frac{2\pi - C}{B} - \left(-\frac{C}{B}\right) = \frac{2\pi}{B}$.

  • Phase: The constant $C$ in $y = A \sin(Bx + C)$ or $y = A \cos(Bx + C)$ is designated as the phase.

  • Physical Application: Phase is significant in field applications such as alternating electrical currents where two sinusoidal waves are compared. If $x$ represents time, phase denotes the fraction of a standard period $2\pi$ that a point on one graph lags or leads a corresponding point on another.

• Combined Transformation Example:

  • For $y = 3 - 5 \sin\left(x + \frac{\pi}{4}\right)$, vertical shift is $k = 3$ (up $3$), amplitude is $|-5| = 5$ with reflection across the midline, period is $2\pi$, and horizontal shift is $\frac{\pi}{4}$ units to the left.

Transformations of Tangent, Cotangent, Secant, and Cosecant

• Tangent and Cotangent Transformations:

  • Because the standard unshifted period of tangent and cotangent is $\pi$, the altered period for $y = A \tan(Bx + C)$ or $y = A \cot(Bx + C)$ is:

  • $\text{Period} = \frac{\pi}{B}$

  • Examples:

  • $y = 3 \tan(x)$ over $\left[-\pi, \pi\right]$: Vertical stretch factor of $3$, period remains $\pi$.

  • $y = \frac{1}{2} \cot(-2x)$: Rewrite using odd symmetry as $y = -\frac{1}{2} \cot(2x)$. Period is $\frac{\pi}{2}$, vertical stretch factor is $\frac{1}{2}$, reflected across the x-axis.

• Secant and Cosecant Transformations:

  • Secant and cosecant curves are graphed by using their reciprocal guide functions (cosine and sine respectively).

  • The period, horizontal translation (phase shift), and vertical translation for secant and cosecant functions are identical to those for sine and cosine.

  • Examples:

  • $y = 4 \csc(x)$: Guide function is $y = 4 \sin(x)$. Graph has asymptotes at $x = k\pi$, relative minimums at $y = 4$, and relative maximums at $y = -4$.

  • $y = -1 - 3 \csc\left(\frac{\pi x}{3} + \frac{\pi}{2}\right)$: Guide function is $y = -1 - 3 \sin\left(\frac{\pi x}{3} + \frac{\pi}{2}\right)$.

    • Period $= \frac{2\pi}{\frac{\pi}{3}} = 6$

    • Phase shift: $\frac{\pi x}{3} + \frac{\pi}{2} = 0 \implies x = -\frac{3}{2}$

    • Midline $y = -1$

Inverse Trigonometric Functions

• Need for Restricted Domains:

  • The graphs of all six trigonometric functions repeat infinitely and fail the horizontal line test across their full domains. Consequently, their unrestricted inverses are non-functional relations.

  • To define inverse relations that are true functions, the domains of the original trigonometric functions must be restricted to intervals where they are one-to-one (passing the horizontal line test) while preserving the full range of the function.

• The Inverse Sine Function ($y = \sin^{-1}(x)$ or $y = \arcsin(x)$):

  • Reflection: Reflecting $y = \sin(x)$ across the line $y = x$ gives the relation $x = \sin(y)$.

  • Restricted Interval: The domain of $y = \sin(x)$ is restricted to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, maintaining the complete range $[-1, 1]$.

  • Definition: The equation $x = \sin(y)$ combined with $\left[-\frac{\pi}{2} \le y \le \frac{\pi}{2}\right]$ defines the inverse sine function.

  • Domain: $[-1, 1]$

  • Range: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

  • Output Angles: Returns angles in Quadrant IV ($y \in \left[-\frac{\pi}{2}, 0\right)$) or Quadrant I ($y \in \left[0, \frac{\pi}{2}\right]$).

• The Inverse Cosine Function ($y = \cos^{-1}(x)$ or $y = \arccos(x)$):

  • Restricted Interval: The domain of $y = \cos(x)$ is restricted to $[0, \pi]$, maintaining the complete range $[-1, 1]$.

  • Definition: The equation $x = \cos(y)$ combined with $[0 \le y \le \pi]$ defines the inverse cosine function.

  • Domain: $[-1, 1]$

  • Range: $[0, \pi]$

  • Output Angles: Returns angles in Quadrant I ($y \in \left[0, \frac{\pi}{2}\right]$) or Quadrant II ($y \in \left(\frac{\pi}{2}, \pi\right]$).

• The Inverse Tangent Function ($y = \tan^{-1}(x)$ or $y = \arctan(x)$):

  • Restricted Interval: The domain of $y = \tan(x)$ is restricted to $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

  • Definition: The equation $x = \tan(y)$ combined with $\left(-\frac{\pi}{2} < y < \frac{\pi}{2}\right)$ defines the inverse tangent function.

  • Domain: All real numbers $(-\infty, \infty)$

  • Range: $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

  • Output Angles: Returns angles in Quadrant IV ($y \in \left(-\frac{\pi}{2}, 0\right)$) or Quadrant I ($y \in \left[0, \frac{\pi}{2}\right)$).

• Exact Evaluation and Simplification Examples:

  • Exact Values (Without Calculator):

  • $\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$

  • $\arccos\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}$

  • $\tan^{-1}(-1) = -\frac{\pi}{4}$

  • Evaluations and Trigonometric Simplification:

  • $\sin\left(\sin^{-1}\left(\frac{1}{2}\right)\right) = \frac{1}{2}$

  • $\sin(\sin(135^\circ)) = \sin\left(\frac{\sqrt{2}}{2}\right)$

  • Simplifying $3 \sec(\theta)$ where $\theta = \tan^{-1}\left(\frac{x}{3}\right)$:

    • Set $\tan(\theta) = \frac{x}{3} = \frac{\text{opposite}}{\text{adjacent}}$

    • By Pythagorean theorem, $\text{hypotenuse} = \sqrt{x^2 + 3^2} = \sqrt{x^2 + 9}$

    • $\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{\sqrt{x^2 + 9}}{3}$

    • Substituting back: $3 \sec(\theta) = 3 \left(\frac{\sqrt{x^2 + 9}}{3}\right) = \sqrt{x^2 + 9}$

  • Converting $\sin(\cos^{-1}(x))$ into an Algebraic Expression in $x$:

    • Let $\theta = \cos^{-1}(x)$, so $\cos(\theta) = x = \frac{x}{1} = \frac{\text{adjacent}}{\text{hypotenuse}}$

    • $\text{opposite} = \sqrt{1 - x^2}$

    • Therefore, $\sin(\cos^{-1}(x)) = \sin(\theta) = \sqrt{1 - x^2}$