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Chapter 4: Problem 90

What is the amplitude of the sine function? What does this tell you about the graph?

Short Answer

Expert verified
The amplitude of a standard sine function is 1, which means that in its graph, it reaches a maximum height of 1 and a minimum height of -1.

Step by step solution

01

Identify a Sine Function

A sine function is normally in the form \(y = a \sin(bx)\), where a is the amplitude and b affects the period of the function. For a standard sine function, such as \(y = \sin(x)\), the amplitude is 1.
02

Understand the Amplitude

The 'amplitude' of a sine function is the distance from the middle line of the graph (the x-axis in the case of the standard sine function) to the peak of each cycle. If \(y = a \sin(bx)\), the amplitude would be the absolute value of a. In case of \(y = \sin(x)\), the amplitude is 1.
03

Relate the Amplitude to the Graph

The amplitude affects the height of the peaks and troughs of the sine function. The higher the amplitude, the taller the peaks and deeper the troughs. For standard sine function \(y = \sin(x)\), it means that the graph will oscillate between -1 and 1 on the vertical y-axis, reaching its peak at \(y = 1\) and its trough at \(y = -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Function
The sine function is a fundamental building block in the realm of trigonometric functions. Typically, it is expressed in the form \(y = a \sin(bx)\). This formula allows for a wide range of transformations based on the values of \(a\) and \(b\).

Here are some key points about the sine function:
  • When \(a = 1\) and \(b = 1\), we get the basic sine function: \(y = \sin(x)\).
  • The function is periodic, meaning it repeats at regular intervals.
  • The range of the sine function is always between -1 and 1, meaning it can never have values outside of this interval.
Understanding the sine function is crucial for exploring other complex trigonometric functions and waves seen in various scientific and engineering applications.
Trigonometric Functions
Trigonometric functions are a set of functions that are highly important in mathematics, particularly in the field of geometry and the study of periodic phenomena. Besides the sine function, other primary trigonometric functions include cosine and tangent.

Some features of trigonometric functions include:
  • They relate angles to ratios of two sides in a right-angle triangle.
  • These functions are periodic, repeating every \(2\pi\) for sine and cosine and \(\pi\) for tangent.
  • Trigonometric functions are used to model oscillations, such as sound waves, light waves, and tides.
The properties and applications make trigonometric functions incredibly versatile in both theoretical and practical scenarios.
Graph of Sine Function
Visualizing the sine function on a graph helps in understanding its wave-like properties. The graph of \(y = \sin(x)\) displays a smooth, continuous wave.

Key characteristics of the sine graph include:
  • The curve rises to a maximum of 1 and falls to a minimum of -1, which corresponds to the amplitude of the sine function.
  • The graph crosses the x-axis at integer multiples of \(\pi\), indicating the function's zero points where \(\sin(x) = 0\).
  • It has a regular pattern, repeating every \(2\pi\) radians, demonstrating its periodicity.
  • The distance from the middle of the wave (usually the x-axis) to the peak is known as the amplitude, and it indicates the "height" of the waves.
These characteristics enable the sine graph to effectively represent many natural phenomena, including vibrations and alternating current in electrical engineering.

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Most popular questions from this chapter

Use words (not an equation) to describe one of the reciprocal identities.

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