Question

Use the graphs in Figures \(4.1\) and \(4.2\) to answer the following questions: What is the maximum possible domain of the function \(y=\sin x\) ?

Short answer

Answer: The maximum possible domain of the function \(y = \sin x\) is the set of all real numbers, expressed as \((-\infty, \infty)\).

Step-by-step solution

Recall Sine function behavior

The sine function is periodic and oscillates between -1 and 1. It completes a full period in an interval of \(2\pi\), which means that every \(2\pi\), the function repeats its values.

Determine the domain

The sine function is defined for all real numbers, since it repeats its values every \(2\pi\). Therefore, the maximum possible domain for the function \(y = \sin x\) is all real numbers.

Give the answer

The maximum possible domain of the function \(y = \sin x\) is the set of all real numbers. In interval notation, this can be expressed as \((-\infty, \infty)\).

Key concepts

Understanding Trigonometric Functions

Trigonometric functions, such as the sine function, are foundational concepts in mathematics that tie angles to ratios of sides in right triangles. As we expand their use beyond right triangles, trigonometric functions become tools for describing waves, oscillations, and circular motion. The sine function, depicted as sin x, is important in fields ranging from physics to engineering and beyond.

Imagine swinging a pendulum: the position of the pendulum over time can be modeled by a sine function. Similarly, sound waves and even the alternating current in your home electricity are described using this mathematical concept. The sine function's connection to the unit circle allows us to associate each real number x (an angle measured in radians) with a point on the circle, and by doing so, define a unique sine value for that angle. This relationship is what allows the sine function to take on every real number as input, explaining its infinitely large domain.

Periodicity of the Sine Function

The periodicity of the sine function is a fundamental property that significantly shapes its graph and applications. In simple terms, periodicity refers to the characteristic of a function where its values repeat at regular intervals. For the sine function, this interval is 2π radians or 360 degrees.

Understanding this repetition is key for many real-world applications. Imagine a Ferris wheel rotating at a constant speed; the height of a seat above the ground varies periodically, much like the sine function. Or consider a heartbeat monitor, where the periodic peaks represent the rhythmic beating of a heart. The sine wave's predictability and continuous nature make it the go-to model for any phenomenon that repeats itself in a consistent cycle.

Interval Notation

Interval notation is a system of writing the set of all numbers between two endpoints, which is incredibly useful in describing domains and ranges of functions. For the sine function, its domain—the set of all possible input values—can be described using interval notation as \( (-\text\backslashinfty, \text\backslashinfty) \). This means that you can input any number from negative infinity to positive infinity into the sine function, and it will yield a valid result.

The beauty of interval notation lies in its compactness and clarity. It's a universal language for expressing how wide or narrow the scope of our mathematical functions is. Whether we are graphing or solving equations, interval notation helps us neatly encapsulate the breadth of possibilities in our mathematical universe.