3.7 Trigonometric Functions

1. Baseline Graphs of Sine, Cosine, and Tangent

Plotting elementary trigonometric outputs against continuous independent inputs generates cyclical geometric waveforms.

The Sine Function ($f(x) = \sin x$):

Domain evaluates to all real numbers ($x \in \mathbb{R}$). Range is strictly bounded between $y \in [-1, 1]$.
The geometric center line tracks identically along the x-axis ($y = 0$).
The maximum amplitude deviation equates to $1$.
The spatial cycle repeats continually establishing a core period of $2\pi$.

The Cosine Function ($f(x) = \cos x$):

Maintains identical domain constraints, boundaries, central axes, amplitude metrics, and periodic thresholds as the sine function.
Geometrically, the cosine waveform is an exact horizontal shift of the sine wave by a displacement of $\dfrac{\pi}{2}$.

The Tangent Function ($f(x) = \tan x$):

The output range is infinitely expansive ($y \in \mathbb{R}$). Amplitude conceptually ceases to exist.
The domain suffers from absolute restrictions at points calculating mathematically undefined. Vertical asymptotes emerge cyclically at $x = \pm\dfrac{\pi}{2}, \pm\dfrac{3\pi}{2}$, etc.
The fundamental repeating cycle accelerates sharply, dictating a shortened period of exactly $\pi$.

2. General Functional Transformations

Complex trigonometric waves are systematically engineered through mathematical adjustments applied to the base function architecture.

The universal structural format $f(x) = A\sin(B(x-D)) + C$ (identically modeled for cosine) dictates strict graphical alterations:

Central Value ($C$): Represents absolute vertical translation. The new geometric central axis traces $y = C$.
Amplitude ($|A|$): Denotes the vertical magnitude scaling factor from the central line. Peak crests are defined by $C + |A|$ and extreme troughs map to $C - |A|$.
Period Modifier ($B$): Enforces horizontal spatial compression. The newly calculated period evaluates mechanically as $T = \dfrac{2\pi}{B}$. Inversely, parameter $B$ extracts via $B = \dfrac{2\pi}{T}$.
Horizontal Shift ($D$): Tracks lateral geometric displacement to the right by $D$ units.

Note: Formulations involving tangents, such as $f(x) = A\tan(B(x-D)) + C$, utilize modified period arithmetic restricted solely to $T = \dfrac{\pi}{B}$. Amplitude technically fails, representing merely a steepness scaling factor.

Extracting Parameters from Graphs

Analyzing physical graphical properties effectively reverse-engineers the mathematical equations determining their shape.

EXAMPLE 1

A sine-style wave cycles visibly between a lowest depth of $10$ and highest peak of $20$, completing a repetition spanning an x-distance of $8$.

The geometric midpoint traces exactly at: $C = \dfrac{20 + 10}{2} \implies \mathbf{C = 15}$.

The physical amplitude scales $5$ units upward from the center: $A = 20 - 15 \implies \mathbf{A = 5}$.

The measured cyclic period translates to $B$: $B = \dfrac{2\pi}{8} \implies \mathbf{B = \dfrac{\pi}{4}}$.

Thus, equation structure dictates: $\mathbf{f(x) = 5\sin\left(\dfrac{\pi}{4}x\right) + 15}$.

EXAMPLE 2

A cosine-style sequence fluctuates continuously from minimum boundary $-10$ to maximum boundary $+20$, fulfilling a phase cycle spanning a total horizontal distance of $\pi$.

Evaluating central equilibrium computes: $C = \dfrac{20 + (-10)}{2} \implies \mathbf{C = 5}$.

Extracting physical variance metrics equates amplitude to: $A = 20 - 5 \implies \mathbf{A = 15}$.

Computing temporal properties generates a B-parameter evaluated as: $B = \dfrac{2\pi}{\pi} \implies \mathbf{B = 2}$.

Thus, equation structure compiles cleanly to: $\mathbf{f(x) = 15\cos(2x) + 5}$.

Determining Optimal Trigonometric Types

A static wave shape can map mathematically to multiple functional definitions simply by shifting the horizontal displacement marker ($D$) to varied functional start points:

Starting geometrically upon the central axis moving upwards matches a standard $\mathbf{+\sin x}$ format.
Starting geometrically upon the central axis plunging downwards matches an inverted $\mathbf{-\sin x}$ format.
Starting geometrically precisely at the maximal crest peak signifies a standard $\mathbf{+\cos x}$ format.
Starting geometrically firmly at the absolute lowest trough assigns an inverted $\mathbf{-\cos x}$ format.

EXAMPLE 4 (Constructing a Target Graph)

Plot the mathematically defined geometry for $f(x) = 5\sin(2x) + 7$ constrained firmly between boundaries $0 \le x \le 2\pi$.

Establishing vertical framework boundaries limits highest points to $12$ and lowest limits to $2$, anchored physically around the central axis $y = 7$.

The cyclic period calculation resolves efficiently to $T = \dfrac{2\pi}{2} = \pi$. Drawing an expanse up to $2\pi$ requires plotting two complete repetitive cycles.

Initiate the geometric line directly atop the central axis, trending steeply upwards to establish proper sine formatting.