Trigonometric functions and graphs Knowledge-Based Study

Study notes

1. Periodic Motion and Trigonometric Functions

   A function $f(x)$ is called periodic if there exists a positive real number $T$ such that $f(x + T) = f(x)$ for all $x$ in the domain. The smallest such positive value $T$ is called the fundamental period of the function.

   Trigonometric functions arise naturally from wrapping the real line around the unit circle defined by $x^2 + y^2 = 1$. For any angle $\theta$ in standard position:

   • The horizontal coordinate represents the cosine function: $x = \cos(\theta)$.
   • The vertical coordinate represents the sine function: $y = \sin(\theta)$.

   Because completing one full revolution around the unit circle corresponds to an angle of $2\pi$ radians, the sine and cosine functions repeat their values every $2\pi$ units. Consequently, both $\sin(x)$ and $\cos(x)$ have a fundamental period of $2\pi$.

   The Fundamental Waves

   The standard sine curve $y = \sin(x)$ and cosine curve $y = \cos(x)$ exhibit continuous, smooth wave patterns over the entire real line.

   Key Properties of $y = \sin(x)$

   • Domain: $(-\infty, \infty)$
   • Range: $[-1, 1]$
   • Period: $2\pi$
   • Symmetry: Odd function, satisfying $\sin(-x) = -\sin(x)$; symmetric about the origin.
   • Key points on $[0, 2\pi]$: $(0, 0)$, $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, and $(2\pi, 0)$.

   Key Properties of $y = \cos(x)$

   • Domain: $(-\infty, \infty)$
   • Range: $[-1, 1]$
   • Period: $2\pi$
   • Symmetry: Even function, satisfying $\cos(-x) = \cos(x)$; symmetric with respect to the $y$-axis.
   • Key points on $[0, 2\pi]$: $(0, 1)$, $(\frac{\pi}{2}, 0)$, $(\pi, -1)$, $(\frac{3\pi}{2}, 0)$, and $(2\pi, 1)$.

   Notice that $\cos(x) = \sin(x + \frac{\pi}{2})$. Thus, the graph of the cosine function is simply the sine wave translated horizontally to the left by $\frac{\pi}{2}$ radians.

   General Sinusoidal Models

   General sinusoidal waves can be expressed in the form:

   $$y = A \sin(B(x - C)) + D \quad \text{or} \quad y = A \cos(B(x - C)) + D$$

   where $A, B, C,$ and $D$ are real constants with $A \neq 0$ and $B > 0$. Each parameter controls a specific geometric transformation:

   • Amplitude ($|A|$): Half the distance between the maximum and minimum values:$$\text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} = |A|$$
   • Period ($T$): The horizontal length of one complete cycle:$$T = \frac{2\pi}{B}$$
   • Phase Shift ($C$): The horizontal shift of the curve. If $C > 0$, the shift is to the right; if $C < 0$, the shift is to the left.
   • Vertical Shift / Midline ($D$): The horizontal axis of symmetry is the line $y = D$, halfway between the peak and trough:$$D = \frac{\text{Maximum} + \text{Minimum}}{2}$$

   Functions with Vertical Asymptotes

   Unlike sine and cosine, the tangent and cotangent functions are defined as ratios:

   $$\tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \cot(x) = \frac{\cos(x)}{\sin(x)}$$

   Because their denominators can equal zero, these functions have points of discontinuity and vertical asymptotes.

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

   • Domain: All real numbers except $x = \frac{\pi}{2} + k\pi$ for any integer $k$.
   • Vertical Asymptotes: $x = \frac{\pi}{2} + k\pi$, where $\cos(x) = 0$.
   • Range: $(-\infty, \infty)$
   • Fundamental Period: $\pi$ (repeats twice as fast as the sine wave).
   • Symmetry: Odd function ($\tan(-x) = -\tan(x)$).
   • Intercepts: $x = k\pi$ for any integer $k$.

   For transformed functions of the form $y = A \tan(Bx)$, the period is given by:

   $$T = \frac{\pi}{|B|}$$

   Analyzing and Sketching a Transformed Wave

   Consider the function:

   $$f(x) = -3 \cos\left(2x - \frac{\pi}{2}\right) + 1$$

   To identify the graph characteristics systematically, factor the coefficient of $x$ inside the argument:

   $$f(x) = -3 \cos\left(2\left(x - \frac{\pi}{4}\right)\right) + 1$$

   Step 1: Determine Key Parameters

   • Amplitude: $|A| = |-3| = 3$
   • Period: $T = \frac{2\pi}{B} = \frac{2\pi}{2} = \pi$
   • Phase Shift: $C = \frac{\pi}{4}$ (shifted $\frac{\pi}{4}$ units to the right)
   • Vertical Shift (Midline): $y = D = 1$
   • Reflection: Because $A = -3 < 0$, the standard cosine curve is reflected across its midline $y = 1$.

   Step 2: Calculate Extrema

   • Maximum value: $D + |A| = 1 + 3 = 4$
   • Minimum value: $D - |A| = 1 - 3 = -2$

   Step 3: Track Critical Points Over One Cycle

   Divide the period $T = \pi$ into 4 equal sub-intervals of width $\Delta x = \frac{\pi}{4}$, starting at the phase shift $x = \frac{\pi}{4}$:

   1. Start at $x = \frac{\pi}{4}$: $f(\frac{\pi}{4}) = -3(1) + 1 = -2$ (Minimum due to reflection)
   2. Quarter-period at $x = \frac{\pi}{2}$: $f(\frac{\pi}{2}) = -3(0) + 1 = 1$ (Midline intercept)
   3. Half-period at $x = \frac{3\pi}{4}$: $f(\frac{3\pi}{4}) = -3(-1) + 1 = 4$ (Maximum)
   4. Three-quarter period at $x = \pi$: $f(\pi) = -3(0) + 1 = 1$ (Midline intercept)
   5. Full period at $x = \frac{5\pi}{4}$: $f(\frac{5\pi}{4}) = -3(1) + 1 = -2$ (Minimum)

   Connecting these points with a smooth, periodic wave gives the complete graph over the interval $[\frac{\pi}{4}, \frac{5\pi}{4}]$.