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Chapter 4: Problem 15

In Exercises 5-18, find the period and amplitude. \(y\ =\ \frac{5}{3}\ cos\ \frac{4x}{5}\)

Short Answer

Expert verified
The amplitude of the function is \(\frac{5}{3}\) and the period is \(\frac{5\pi}{2}\).

Step by step solution

01

Find the Amplitude

The amplitude 'A' of a cosine function \(y = a \cos(bx)\) is given by the absolute value of 'a'. Here the 'a' is \(\frac{5}{3}\). So, the amplitude A = abs(\(\frac{5}{3}\)) = \(\frac{5}{3}\).
02

Find the Period

The period 'T' of a cosine function \(y = a \cos(bx)\) is found by dividing \(2\pi\) by the absolute value of 'b'. From the given function, 'b' equals \(\frac{4}{5}\). Therefore, the period T = \(\frac{2\pi}{abs(\frac{4}{5})}\) = \(\frac{2\pi}{\frac{4}{5}}\) = \(\frac{5\pi}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
In trigonometric functions like the cosine function, "amplitude" refers to the maximum deviation or distance from the central axis to the peak. This measurement tells you how high or low the function will go relative to the horizontal center line.

Amplitude is always a positive value, calculated as the absolute value of the coefficient in front of the cosine term. For example, in the given function, which is in the form of \(y = a \cos(bx)\), the "a" term \(\frac{5}{3}\) is used to find the amplitude. Thus, the amplitude is \(\frac{5}{3}\).

Some key points to remember about amplitude:
  • It is always a positive number.
  • Determines the height of the wave peaks.
  • Independent of the wave's period or frequency.
Period
The period of a trigonometric function is the interval it takes to complete one full cycle. It helps you understand how frequent the waves occur in the given function. In simpler terms, it is the distance along the x-axis before the pattern of the wave repeats.

To calculate the period for a cosine function like \(y = a \cos(bx)\), you take \(2\pi\) and divide it by the number in front of x, known as "b". In our problem, "b" is \(\frac{4}{5}\). Hence, the period \(T\) is calculated as \(\frac{2\pi}{\frac{4}{5}}\), which simplifies to \(\frac{5\pi}{2}\).

Here鈥檚 a summary about periods:
  • The larger the value of "b", the shorter the period.
  • A shorter period means more cycles in a given interval.
  • Longer periods result in wider waves.
Cosine Function
The cosine function is one of the fundamental trigonometric functions, represented by \(\cos\). It is a periodic wave that oscillates between defined maximum and minimum values. In the standard form \(y = a \cos(bx)\), the parameters "a" and "b" affect the graph's shape and characteristics.

For this function:
  • "a" determines the amplitude, affecting how tall the wave is.
  • "b" influences the period, which is how often waves are completed.
  • The cosine wave is symmetric about the vertical axis and has a range from -amplitude to +amplitude.
Cosine waves are important in various fields such as physics, engineering, and signal processing, as they model cyclic behaviors and oscillations efficiently.

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Most popular questions from this chapter

GEOMETRY In Exercises 43 and 44, find the angle \(\alpha\) between two nonvertical lines \(L_1\) and \(L_2\). The angle \(\alpha\) satisfies the equation \(\tan \alpha =\ \left| \dfrac{m_2 - m_1}{1+ m_2 m_1} \right|\) where \(m_1\) and \(m_2\) are the slopes of \(L_1\) and \(L_2\), respectively. (Assume that \(m_1 m_2 \neq -1\).) \(L_1\): \(2x - y = 8\) \(L_2\): \(x - 5y = -4\)

THINK ABOUT IT Consider the functions given by \(f(x)= \sin\ x\) and \(f^{-1}(x)= \arcsin\ x\). (a) Use a graphing utility to graph the composite functions \(f \circ f^{-1}\) and \(f^{-1} \circ f\). (b) Explain why the graphs in part (a) are not the graph of the line \(y=x\). Why do the graphs of \(f \circ f^{-1}\) and \(f^{-1} \circ f\) differ?

In Exercises 67-76, write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) cot(arctan \(\dfrac{1}{x}\))

TRUE OR FALSE? In Exercises 112-114, determine whether the statement is true or false. Justify your answer. \(\arctan\ x\ =\ \dfrac{\arcsin x}{\arccos x}\)

In Exercises 55-66, find the exact value of the expression. (Hint:Sketch a right triangle.) \(sin[arccos\ (-\frac{2}{3})]\)
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