/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 78 Suppose \(f\) is the function wh... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 78

Suppose \(f\) is the function whose value at \(x\) is the cosine of \(x\) degrees. Explain how the graph of \(f\) is obtained from the graph of \(\cos x\).

Short Answer

Expert verified
To obtain the graph of the function \(f(x) = \cos(x^{\circ})\) from the graph of \(\cos x\), we first convert the input x from degrees to radians using the formula \(x \, radians = x \, degrees \times \frac{\pi}{180}\). This gives us the function \(f(x) = \cos (\frac{\pi x}{180})\). Next, we plot the graph of \(f(x) = \cos (\frac{\pi x}{180})\) using the properties of the cosine function, such as periodicity, evenness, and amplitude. The main differences between the graphs are the units on the x-axis (degrees for \(f(x)\) vs radians for \(\cos x\)) and the period (360 degrees for \(f(x)\) vs \(2\pi\) radians for \(\cos x\)).

Step by step solution

01

Convert function from degrees to radians

Since f(x) = cos(x degrees), we need to convert the input x from degrees to radians. To do so, we use the conversion factor: \(1 radian = \frac{180}{\pi} degrees\). We want x in radians, so we multiply x degrees by the reciprocal of the conversion factor: \(x radians = x degrees \times \frac{\pi}{180}\). Thus, the function in radians is: \(f(x) = \cos \left(\frac{\pi x}{180}\right)\).
02

Identify cosine function properties

The cosine function has some important properties, such as: 1. Periodic: The cosine function has a period of \(2\pi\) radians. This means that \(\cos(x) = \cos(x + 2\pi n)\) for any integer n. 2. Even function: The cosine function is an even function, meaning \(\cos(x) = \cos(-x)\) for all x. 3. Amplitude: The amplitude of the cosine function is 1, which means the maximum height above the x-axis is 1 and the minimum height below the x-axis is also 1.
03

Plot the converted function

Now we can plot the graph of \(f(x) = \cos \left(\frac{\pi x}{180}\right)\) using the properties of the cosine function. Draw the x-axis and label it with degrees. Start with the x values as multiples of 180 degrees and graph the function, keeping in mind the periodic and even properties as well as the amplitude.
04

Compare the graphs

After plotting the graph of \(f(x) = \cos \left(\frac{\pi x}{180}\right)\), we can compare it with the graph of \(\cos x\). The main differences are the units on the x-axis (degrees vs radians) and the period (360 degrees or \(2\pi\) radians for \(\cos x\) vs 360 degrees for f(x)). However, the shape, amplitude, and even properties of the graphs are the same. In conclusion, the main difference in obtaining the graph of f(x) from the graph of cos(x) is the conversion of x from degrees to radians and adjusting the x-axis units accordingly.

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