Glossary

4.1 Graphing Sine and Cosine Functions

2 min read•july 25, 2024

Sine and cosine functions are the backbone of trigonometry. They create smooth, repeating waves that oscillate between -1 and 1. These functions help us model real-world phenomena like sound waves and planetary orbits.

Understanding how to graph sine and cosine is crucial. Key points, intercepts, and periodicity are essential concepts. Knowing the domain, range, and how transformations affect these graphs will give you a solid foundation for more advanced trigonometric applications.

Graphing Sine and Cosine Functions

Graphing sine and cosine functions

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  • Key points for sine function plot smooth curve through (0, 0), (π2\frac{\pi}{2}, 1), (π\pi, 0), (3π2\frac{3\pi}{2}, -1), (2π2\pi, 0)
  • Key points for cosine function plot smooth curve through (0, 1), (π2\frac{\pi}{2}, 0), (π\pi, -1), (3π2\frac{3\pi}{2}, 0), (2π2\pi, 1)
  • Intervals one complete cycle [0,2π][0, 2\pi] or [−π,π][-\pi, \pi], quarter-cycle intervals [0,π2][0, \frac{\pi}{2}], [π2,π][\frac{\pi}{2}, \pi], [π,3π2][\pi, \frac{3\pi}{2}], [3π2,2π][\frac{3\pi}{2}, 2\pi]
  • distance from midline to max or min point 1 for standard sine and cosine functions
  • length of one complete cycle 2π2\pi for standard sine and cosine functions
  • Sine and cosine graphs related by horizontal shift of π2\frac{\pi}{2}

Domain and range of trigonometric functions

  • Domain all real numbers (-∞, ∞) in interval notation for both sine and cosine
  • Range [-1, 1] for both sine and cosine functions limited by amplitude
  • Continuous functions no breaks or gaps in graph smooth curve throughout domain
  • Sine function oscillates between -1 and 1, passing through 0
  • Cosine function oscillates between -1 and 1, starting at 1

Intercepts of sine and cosine

  • Sine x-intercepts occur at multiples of π\pi (0, π\pi, 2π2\pi, ...)
  • Cosine x-intercepts occur at odd multiples of π2\frac{\pi}{2} (π2\frac{\pi}{2}, 3π2\frac{3\pi}{2}, ...)
  • Sine y-intercept (0, 0) function passes through origin
  • Cosine y-intercept (0, 1) function starts at maximum value
  • General formulas for x-intercepts:
    1. Sine: x=nπx = n\pi, n is any integer
    2. Cosine: x=π2+nπx = \frac{\pi}{2} + n\pi, n is any integer

Periodicity in trigonometric functions

  • Periodicity function repeats values at regular intervals (sound waves)
  • Period length of one complete cycle 2π2\pi for standard sine and cosine
  • number of cycles completed in given interval inverse of period
  • Transformations affecting periodicity:
    • Horizontal stretch/compression changes period
    • Vertical stretch/compression does not affect period
  • Applications model cyclical phenomena (planetary orbits, tides, alternating current)

Key Terms to Review (16)

0 degrees: 0 degrees is a crucial angle in trigonometry that serves as the starting point on the unit circle, representing the position of a point at the intersection of the positive x-axis. This angle plays a significant role in defining the sine and cosine functions, as it helps establish their values at key intervals. Understanding 0 degrees is essential for graphing sine and cosine functions because it sets the baseline for their behavior and periodicity.
180 degrees: 180 degrees represents a critical angle in trigonometry, specifically marking a half turn in a circle. It is essential in understanding the periodic nature of sine and cosine functions, as it indicates where these functions exhibit symmetry and phase shifts. This angle serves as a reference point for determining the coordinates of points on the unit circle, which are pivotal when graphing sine and cosine functions.
360 degrees: 360 degrees is a complete rotation measurement that describes the total angle around a point. In the context of trigonometric functions, particularly sine and cosine, this measurement represents the full cycle of periodic functions, where one complete rotation can be illustrated on the unit circle. Understanding how 360 degrees relates to sine and cosine helps in graphing these functions and analyzing their properties effectively.
90 degrees: 90 degrees is a specific angle measurement that is fundamental in trigonometry and represents a quarter of a complete rotation or one-fourth of a circle. This angle plays a critical role in defining the properties and behaviors of sine and cosine functions, which are crucial in analyzing periodic phenomena, and it serves as a reference point for evaluating various trigonometric functions, identities, and equations.
Amplitude: Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position in trigonometric functions, particularly in sine and cosine graphs. This term is crucial when understanding how these functions oscillate and can be transformed, impacting their height and overall shape. It plays a significant role in determining the vertical stretch or compression of the graph, influencing how the wave interacts with shifts and periodicity.
Coordinate Plane: A coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis), where each point is identified by a pair of numerical coordinates. This system allows for the graphical representation of mathematical functions, including sine and cosine functions, providing a visual context for understanding their behavior and properties.
Even Function: An even function is a type of function where the output value remains unchanged when the input value is negated. This means that for every value of $$x$$, the equation $$f(x) = f(-x)$$ holds true. In the context of graphing, even functions exhibit symmetry about the y-axis, making them visually distinctive and easily recognizable.
Frequency: Frequency refers to the number of complete cycles of a periodic function that occur in a unit of time. In trigonometry, it's particularly relevant when analyzing how often a sine or cosine wave oscillates, which can be influenced by transformations like stretching or compressing the graph. Understanding frequency helps in solving real-world problems where periodic behavior, such as sound waves or seasonal changes, is involved.
Odd Function: An odd function is a type of mathematical function that exhibits symmetry about the origin, meaning that if you take any point (x, y) on the graph of the function, the point (-x, -y) will also be on the graph. This property leads to specific behaviors in functions like sine and cosine when graphed, and it plays a significant role in understanding their periodic nature and transformations.
Oscillation: Oscillation refers to the repetitive back-and-forth movement of an object or a wave, often described in the context of periodic functions such as sine and cosine. This term is fundamental in understanding wave patterns, where it illustrates how values rise and fall over time, impacting the amplitude, period, and overall shape of these functions. In mathematical graphs, oscillation visually represents cycles of motion, creating the smooth curves that depict phenomena in various fields such as physics and engineering.
Period: In trigonometry, the period is the length of one complete cycle of a periodic function, which is essential for understanding the behavior of functions like sine, cosine, tangent, and their transformations. The period determines how often a function repeats its values, allowing us to analyze and predict the function's behavior across its domain. Knowing the period helps in visualizing the graph and understanding the effects of transformations, such as stretching or compressing.
Phase Shift: Phase shift refers to the horizontal displacement of a trigonometric graph from its standard position. It indicates how much the graph of a sine or cosine function is shifted left or right along the x-axis, which is crucial for accurately representing real-world scenarios. Understanding phase shift helps in determining how periodic phenomena can be adjusted in terms of their timing and alignment with other functions.
Unit circle: The unit circle is a circle with a radius of one centered at the origin of a coordinate system, commonly used to define trigonometric functions. It serves as a fundamental tool for visualizing the relationships between angles and their corresponding sine and cosine values, helping to extend trigonometric functions to all real numbers and providing insights into complex numbers and their roots.
Vertical Shift: A vertical shift refers to the transformation of a trigonometric graph that moves it up or down along the y-axis. This adjustment alters the graph's midline without changing its shape, amplitude, or period. Understanding vertical shifts is crucial as they impact how functions behave and interact within various contexts, including periodic behavior and function transformations.
Waveform: A waveform is a graphical representation of how a signal varies over time, typically illustrating amplitude, frequency, and phase. It is crucial in understanding the behavior of periodic functions, especially sine and cosine functions, as these waveforms depict oscillations that repeat at regular intervals. Key characteristics like amplitude and period can be derived from waveforms, showcasing their relationship with trigonometric concepts.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, meaning that at this point, the value of y is zero. Identifying x-intercepts is crucial for understanding the behavior of functions, particularly in determining where functions change direction and how they relate to their roots. This concept is fundamental when analyzing periodic functions and their graphs, as it helps in identifying key points that define the shape and movement of the waveforms.
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