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Chapter 4: Problem 88

What is the range of the sine function? Use the unit circle to explain where this range comes from.

Short Answer

Expert verified
The range of the sine function is [-1, 1]

Step by step solution

01

Understand the Representation of the Sine Function

In the unit circle, the sine function is represented by the y-coordinate of a point on the circumference. As the angle progresses from 0 to \(2\pi\) (or 0 to 360 degrees), the y-coordinate oscillates between -1 and 1 inclusive.
02

Identify Maximum and Minimum Values

The maximum y-value is achieved at \(\pi/2\) or 90 degrees and the minimum y-value is observed at \(3\pi/2\) or 270 degrees. These maximum and minimum are 1 and -1 respectively.
03

Define the Range

Having identified the maximum and minimum values of the sine function, the range is therefore all values that the function can take on. The range of the sine function is therefore the closed interval from -1 to 1 inclusive. This is denoted as [-1, 1]

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Most popular questions from this chapter

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-2.5 \sin \frac{\pi}{3} x \text { and } y=-2.5 \csc \frac{\pi}{3} x$$

Find the slant asymptote of $$ f(x)=\frac{2 x^{2}-7 x-1}{x-2} $$ (Section \(2.6, \text { Example } 8)\)

Graph one period of each function. $$y=\left|3 \cos \frac{2 x}{3}\right|$$

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=-3 \cos \left(2 x-\frac{\pi}{2}\right)$$

Find all zeros of \(f(x)=2 x^{3}-5 x^{2}+x+2\) (Section \(2.5, \text { Example } 3)\)
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