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Chapter 4: Problem 89

Without drawing a graph, describe the behavior of the basic sine curve.

Short Answer

Expert verified
The basic sine curve is a smooth, repetitive oscillation starting from the origin (0,0). It has a wavelength of \(2\pi\), meaning it repeats every \(2\pi\) units and an amplitude of 1, meaning it reaches a maximum height of 1 and a minimum height of -1. The curve crosses the x-axis at \(x = n\pi\) for all integers n.

Step by step solution

01

Define the Sine Function

The sine function, often denoted as sin(x), is a periodic function that describes a smooth, repetitive oscillation. It鈥檚 one of the basic functions in trigonometry. The standard sine curve y = sin(x) has a wavelength (the length of one whole wave) of \(2\pi\), an amplitude (the height from the middle to the top or bottom) of 1.
02

Explain the Key Points on the Sine Curve

The sine curve begins at the origin (0,0). As x increases, it goes up to a maximum height of 1 at \(x = \frac{\pi}{2}\), then comes down again and reaches the minimum height of -1 at \(x = \frac{3\pi}{2}\) before returning to 0 at \(x = 2\pi\). This pattern then repeats itself for all larger x-values.
03

Describe the Periodicity of the Sine Curve

The sine curve is periodic, meaning it repeats its pattern every \(2\pi\) units. This is called the period of the sine function. For all integer values of n, the curve has x-intercepts (where the curve crosses the x-axis) at \(x = n\pi\). Moreover, for the sine function, the maximum amplitude is 1, and the minimum amplitude is -1.

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