Sine and cosine functions are the backbone of periodic modeling. These versatile tools help us understand and predict cyclical patterns in nature, from tides to temperature changes. Their graphs share key features like , , and phase shift.
Mastering these functions opens doors to real-world applications. By tweaking parameters, we can model complex periodic behaviors. Understanding the relationship between sine and cosine graphs also helps us tackle a wide range of mathematical and scientific problems.
Graphs of Sine and Cosine Functions
Key features of sine and cosine graphs
General form of : y=asin(b(x−c))+d
a represents amplitude determines vertical stretch or compression of the graph
b affects period calculated by the formula ∣b∣2π
c represents phase shift causes horizontal translation of the graph
d represents moves the graph up or down
General form of : y=acos(b(x−c))+d
Parameters a, b, c, and d have the same effects as in the sine function
Amplitude measures distance between the and the maximum or minimum value of the function
Determined by the absolute value of the coefficient a
Example: If a=2, the amplitude is 2 units above and below the midline
Period represents length of one complete cycle of the function
Determined by ∣b∣2π, where b is the coefficient of x inside the sine or cosine function
Example: If b=4π, the period is 4π2π=8 units
Phase shift describes horizontal translation of the graph
Determined by the value of c inside the sine or cosine function
Positive c values shift the graph to the right, while negative values shift it to the left
Example: If c=2π, the graph shifts 2π units to the right
Frequency is the number of cycles completed in a given time period (reciprocal of the period)
Transformations for periodic modeling
Identify the periodic behavior in the real-world situation (temperature, tides, sound waves)
Determine the baseline (midline) of the
Represented by the vertical shift parameter d
Example: Average daily temperature over a year
Determine the amplitude of the periodic function
Represented by the absolute value of the coefficient a
Example: Maximum deviation from the average temperature
Determine the period of the periodic function
Use the formula ∣b∣2π, where b is the coefficient of x inside the sine or cosine function
Example: Length of one complete cycle (day, month, year)
Determine any phase shift (horizontal translation) of the periodic function
Represented by the value of c inside the sine or cosine function
Example: Time delay between the start of the cycle and the occurrence of the maximum value
Construct the appropriate sine or cosine function using the determined parameters
Consider if the situation represents harmonic motion, which can be modeled using trigonometric functions
Relationships between sine and cosine
Sine and cosine graphs are horizontally translated versions of each other
sin(x)=cos(x−2π)
cos(x)=sin(x+2π)
Both sine and cosine graphs have symmetry about the midline
Sine graph symmetric about the origin
Cosine graph symmetric about the y-axis
Horizontal translations between sine and cosine graphs are always 2π units (90 degrees)
Shifting a sine graph to the left by 2π units results in a cosine graph
Shifting a cosine graph to the right by 2π units results in a sine graph
Wave Functions and Periodic Behavior
Wave functions are mathematical descriptions of waves, often represented by trigonometric functions
Periodic functions repeat their values at regular intervals, with sine and cosine being common examples
Angular frequency (ω) describes how quickly the function oscillates, related to the period by ω = 2π/T, where T is the period
Key Terms to Review (10)
Amplitude: Amplitude is the maximum absolute value of a periodic function measured from its average or equilibrium position. It represents the height of the wave peaks or troughs from the center line.
Circular motion: Circular motion describes the movement of an object along the circumference of a circle. It is characterized by a constant distance from a fixed central point.
Cosine function: The cosine function, denoted as $\cos(\theta)$, is a trigonometric function that represents the x-coordinate of a point on the unit circle at an angle $\theta$ from the positive x-axis. It is periodic with a period of $2\pi$ and ranges from -1 to 1.
Horizontal shift: A horizontal shift is a transformation that moves a graph left or right along the x-axis without changing its shape. It is represented by modifying the function as $f(x) \rightarrow f(x - h)$, where $h$ is the number of units shifted.
Midline: The midline of a periodic function is the horizontal axis that runs through the middle of the graph, equidistant from its maximum and minimum values. It represents the average value of the function over one period.
Period: The period of a trigonometric function is the interval over which it completes one full cycle and starts to repeat. For sine and cosine functions, the period is $2\pi$.
Periodic function: A periodic function is a function that repeats its values in regular intervals or periods. The most common examples are the sine and cosine functions.
Sine function: The sine function, denoted as $\sin(\theta)$, relates the angle $\theta$ in a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is a periodic function with a period of $2\pi$.
Sinusoidal function: A sinusoidal function is a type of periodic function that describes smooth, repetitive oscillations, typically modeled using the sine or cosine functions. These functions are fundamental in trigonometry and have applications in various fields such as physics, engineering, and signal processing.
Vertical shift: A vertical shift is a transformation that moves a graph up or down by adding or subtracting a constant to the function's output. This does not change the shape of the graph, only its position along the y-axis.