8.1 Graphs of the Sine and Cosine Functions

Sine and cosine functions are the building blocks of periodic motion. These waves show up everywhere, from sound to light to ocean tides. Understanding how to graph and manipulate them is key to modeling real-world phenomena.

The notes cover how to adjust sine and cosine graphs using amplitude, frequency, phase shift, and vertical shift. They also explain how to identify these elements from a graph and write equations. This knowledge is crucial for analyzing and predicting cyclic behavior.

Graphs of Sine and Cosine Functions

Graphing sine and cosine variations

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• General form of sine and cosine functions: $y = A \sin(B(x - C)) + D$ and $y = A \cos(B(x - C)) + D$
  • $A$: Amplitude, determines the vertical stretch or compression of the graph
    • $|A| > 1$ stretches the graph vertically (taller waves)
    • $|A| < 1$ compresses the graph vertically (shorter waves)
    • $A < 0$ reflects the graph over the x-axis (upside-down waves)
  • $B$: Frequency, related to the period and affects the horizontal compression or stretch
    • As $|B|$ increases, the period decreases, compressing the graph horizontally (more waves in the same space)
    • As $|B|$ decreases, the period increases, stretching the graph horizontally (fewer waves in the same space)
    • $B$ is also known as the angular frequency in the context of harmonic motion
  • $C$: Phase shift, represents the horizontal shift of the graph
    • $C > 0$ shifts the graph to the right (delayed start)
    • $C < 0$ shifts the graph to the left (early start)
  • $D$: Vertical shift, moves the graph up or down
    • $D > 0$ shifts the graph up (higher waves)
    • $D < 0$ shifts the graph down (lower waves)
• Period ($P$) is the length of one complete cycle of the function
  • Calculated using the formula $P = \frac{2\pi}{|B|}$, where $B$ is the frequency
  • Examples of periods: $2\pi$ (standard sine or cosine), $\pi$ (double frequency), $4\pi$ (half frequency)
  • Period is measured in radians when using the unit circle

Key features of sinusoidal graphs

• Midline is the horizontal line around which the graph oscillates
  • Determined by the vertical shift ($D$)
  • Equation of the midline: $y = D$
  • Examples of midlines: $y = 0$ (standard sine or cosine), $y = 2$ (shifted up by 2), $y = -1$ (shifted down by 1)
• Amplitude is the maximum distance between the midline and the maximum or minimum points of the graph
  • Determined by the absolute value of $A$
  • Examples of amplitudes: 1 (standard sine or cosine), 2 (stretched vertically), 0.5 (compressed vertically)
• Extrema are the maximum and minimum points of the graph
  • For sine functions:
    1. Maximum occurs at $x = \frac{\pi}{2|B|} + \frac{C}{B} + \frac{2\pi n}{|B|}$
    2. Minimum occurs at $x = -\frac{\pi}{2|B|} + \frac{C}{B} + \frac{2\pi n}{|B|}$
  • For cosine functions:
    1. Maximum occurs at $x = \frac{C}{B} + \frac{2\pi n}{|B|}$
    2. Minimum occurs at $x = \frac{\pi}{|B|} + \frac{C}{B} + \frac{2\pi n}{|B|}$
  • Where $n$ is any integer
  • Examples of extrema: (0, 1) and (π, -1) for standard sine, (0, 1) and (π, 1) for standard cosine

Equations from sinusoidal graphs

• Identify the midline ($D$) from the graph or context
  • The midline is the horizontal line that the graph oscillates around
  • Examples: a tide graph with a midline at the average sea level, a sound wave with a midline at atmospheric pressure
• Determine the amplitude ($A$) by measuring the distance from the midline to the maximum or minimum
  • Examples: the height of a wave from the average water level to the crest or trough, the loudness of a sound from the average to the peak
• Find the period ($P$) by measuring the length of one complete cycle
  • Calculate the frequency using $B = \frac{2\pi}{P}$
  • Examples: the time between high tides (about 12 hours), the time for one complete rotation of a Ferris wheel
• Identify the phase shift ($C$) by comparing the graph to the parent function
  • For sine, find the horizontal distance from the origin to the nearest maximum point
  • For cosine, find the horizontal distance from the origin to the nearest maximum or minimum point
  • Examples: a tide graph starting at high tide (phase shift of π/2 for sine), a Ferris wheel starting at the bottom (phase shift of π for cosine)
• Substitute the values of $A$, $B$, $C$, and $D$ into the general form of the function
  • $y = A \sin(B(x - C)) + D$ for sine functions
  • $y = A \cos(B(x - C)) + D$ for cosine functions
  • Example: a tide graph with an amplitude of 2 m, a period of 12 hours, a phase shift of 3 hours, and a midline at 4 m would have the equation $y = 2 \sin(\frac{\pi}{6}(x - 3)) + 4$

Additional Concepts

• Periodic functions repeat their values at regular intervals, with sine and cosine being common examples
• The unit circle is a fundamental tool for understanding trigonometric functions and their relationships
• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables