Sinusoidal functions are like waves, repeating patterns that show up in nature and math. They have key parts: amplitude (height), period (length), and shifts. These functions help us model real-world stuff like sound waves and tides.

Harmonic motion is a special type of back-and-forth movement, like a swinging pendulum. It's described by similar wave-like equations, but with time as a factor. This motion shows up in physics and helps explain how things vibrate or oscillate.

Sinusoidal Functions

Amplitude and period of sinusoids

• Amplitude
  • Vertical distance between midline and maximum or minimum point of function
  • Determined by absolute value of coefficient of trigonometric function ($3$ for $y = 3\sin(x)$)
• Period
  • Length of one complete cycle of function
  • Calculated using formula $\frac{2\pi}{|b|}$, where $b$ is coefficient of $x$ inside trigonometric function ($\pi$ for $y = \sin(2x)$)

Graphing and equations for sinusoids

• General form of sinusoidal function: $y = A\sin(B(x - C)) + D$ or $y = A\cos(B(x - C)) + D$
  • $A$ represents amplitude
  • $B$ determines period and frequency
  • $C$ represents phase shift (horizontal shift)
  • $D$ represents vertical shift (also known as the midline)
• Graphing steps
  1. Determine amplitude, period, phase shift, and vertical shift from equation
  2. Sketch graph using these parameters
• Creating equations
  1. Identify amplitude, period, phase shift, and vertical shift from given information or graph
  2. Substitute values into general form of sinusoidal function

Modeling with trigonometric functions

• Periodic phenomena are events that repeat at regular intervals (sound waves, tides, seasonal temperatures, daylight hours)
• Steps to model periodic phenomena
  1. Identify amplitude, period, phase shift, and vertical shift of data
  2. Choose appropriate trigonometric function (sine or cosine) based on initial conditions
  3. Create equation using general form of sinusoidal function
  4. Adjust parameters to fit data points
• Wave functions can be used to model various periodic phenomena in nature and physics

Harmonic Motion

Harmonic motion in trigonometry

• Harmonic motion is type of periodic motion where object oscillates about equilibrium position (pendulums, springs, vibrating strings)
• Simple harmonic motion equation: $x(t) = A\cos(\omega t + \phi)$ or $x(t) = A\sin(\omega t + \phi)$
  • $A$ represents amplitude
  • $\omega$ represents angular frequency, related to period by $\omega = \frac{2\pi}{T}$
  • $\phi$ represents phase shift
  • $t$ represents time
• Velocity and acceleration in harmonic motion
  • Velocity: $v(t) = -A\omega\sin(\omega t + \phi)$ (for cosine position function) or $v(t) = A\omega\cos(\omega t + \phi)$ (for sine position function)
  • Acceleration: $a(t) = -A\omega^2\cos(\omega t + \phi)$ (for cosine position function) or $a(t) = -A\omega^2\sin(\omega t + \phi)$ (for sine position function)
• Angular velocity (ω) describes the rate of change of the angle in radians per unit time

Oscillation and Equilibrium

• Oscillation refers to the repetitive variation of a quantity or position about a central value
• Equilibrium position is the resting or center point around which the oscillation occurs