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Chapter 5: Problem 20

Find the period and amplitude. $$y=\frac{5}{2} \cos \frac{x}{4}$$

Short Answer

Expert verified
The amplitude is \( \frac{5}{2}\) and the period is \(8\pi \).

Step by step solution

01

Determine the amplitude

The amplitude of a cosine function is the absolute value of the leading coefficient. In this case, the leading coefficient is \( \frac{5}{2}\), so the amplitude of \(y = \frac{5}{2} \cos \frac{x}{4}\) is \( \frac{5}{2}\).
02

Determine the period

The period of a cosine function is calculated as \(2\pi\) divided by the absolute value of the coefficient of x inside the cosine function. In this case, the coefficient of x is \(\frac{1}{4}\), so the period is \(2\pi \times 4 = 8\pi \).

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Most popular questions from this chapter

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\sec \theta=3\)

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=30^{\circ}$$

True or False Determine whether the statement is true or false. Justify your answer. The tangent function can be used to model harmonic motion.

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The table shows the average daily high temperatures (in degrees Fahrenheit) for Quillayute, Washington, \(Q\) and Chicago, Illinois, \(C\) for month \(t,\) with \(t=1\) corresponding to January. $$\begin{array}{c|c|c} \text { Month, } & \text { Quillayute, } & \text { Chicago, } \\ t & Q & C \\ \hline 1 & 47.1 & 31.0 \\ 2 & 49.1 & 35.3 \\ 3 & 51.4 & 46.6 \\ 4 & 54.8 & 59.0 \\ 5 & 59.5 & 70.0 \\ 6 & 63.1 & 79.7 \\ 7 & 67.4 & 84.1 \\ 8 & 68.6 & 81.9 \\ 9 & 66.2 & 74.8 \\ 10 & 58.2 & 62.3 \\ 11 & 50.3 & 48.2 \\ 12 & 46.0 & 34.8 \end{array}$$ (a) \(\mathrm{A}\) model for the temperature in Quillayute is given by $$Q(t)=57.5+10.6 \sin (0.566 x-2.568)$$ Find a trigonometric model for Chicago. (b) Use a graphing utility to graph the data and the model for the temperatures in Quillayute in the same viewing window. How well does the model fit the data? (c) Use the graphing utility to graph the data and the model for the temperatures in Chicago in the same viewing window. How well does the model fit the data? (d) Use the models to estimate the average daily high temperature in each city. Which term of the models did you use? Explain. (e) What is the period of each model? Are the periods what you expected? Explain. (f) Which city has the greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain.
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