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 $\frac{\pi}{2}$.
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 $x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}$, 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 peaks 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 $T = \frac{2\pi}{B}$. Inversely, parameter $B$ extracts via $B = \frac{2\pi}{T}$.
- 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 $T = \frac{\pi}{B}$. Amplitude technically fails, representing merely a steepness scaling factor.
EXAMPLE 1 & 2 (Extracting Parameters from Graphs)
Analyzing physical graphical properties effectively reverse-engineers the mathematical equations determining their shape.
The geometric midpoint traces exactly at $y = 15$, cementing $\mathbf{C = 15}$.
The physical amplitude scales 5 units upward from the center, securing $\mathbf{A = 5}$.
The measured cyclic period translates to $B$: $B = \frac{2\pi}{8} = \frac{\pi}{4}$.
Thus, equation structure dictates $\mathbf{f(x) = 5\sin\left(\frac{\pi}{4}x\right) + 15}$.
Evaluating central equilibrium computes $C = \frac{20 + (-10)}{2} = \mathbf{5}$.
Extracting physical variance metrics equates amplitude to $A = 20 - 5 = \mathbf{15}$.
Computing temporal properties generates a B-parameter evaluated as $B = \frac{2\pi}{\pi} = \mathbf{2}$.
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 $0 \le x \le 2\pi$.