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Chapter 4: Problem 1

One period of a sine or cosine function is called one _____ of the sine or cosine curve.

Short Answer

Expert verified
The blank should be filled with the word 'period'.

Step by step solution

01

Identify the Question

The exercise is requesting for a term that gives the name of one complete cycle of a sine or cosine curve.
02

Apply Knowledge on Sine and Cosine Curves

A sine or cosine function repeats its values in regular intervals in its domain. This is due to the periodic nature of these functions. One complete cycle of a sine or cosine graph is called a period.
03

Answer the Question

The term that fits the context of the sentence is 'period'. So, one period of a sine or cosine function is referred to as one period of the sine or cosine curve.

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Key Concepts

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

Sine Function
The sine function is a fundamental trigonometric function often denoted as \( \sin(\theta) \). It deals with the ratio of the length of the opposite side to the hypotenuse in a right triangle. This function starts from zero, increases to one, decreases back through zero to negative one, and finally returns to zero, completing a cycle. Trigonometric functions help model periodic phenomena, such as sound waves or tides.
  • Based on an angle \( \theta \), the sine function oscillates between -1 and 1.
  • Its graph, known as the sine wave, smoothly undulates and is symmetric with respect to the origin.
  • Key points on the sine curve occur at \( \theta = 0 \), \( \theta = \frac{\pi}{2} \), \( \theta = \pi \), and \( \theta = \frac{3\pi}{2} \).

The smooth, wave-like shape of the sine curve makes it useful in various fields, from engineering to physics. Remember, the sine function is periodical and repeats its values in regular intervals.
Cosine Function
The cosine function, denoted by \( \cos(\theta) \), is another essential trigonometric function. It's defined as the ratio of the length of the adjacent side to the hypotenuse of a right triangle. The cosine function, like the sine, is periodic and shares similar characteristics with the sine function.
  • The cosine function starts at its maximum value of 1 when \( \theta = 0 \).
  • It then decreases to -1, rises back to 1, completing a cycle or period.
  • This function oscillates similar to the sine function; however, its phase is shifted.
  • Important points for cosine occur at \( \theta = 0 \), \( \theta = \pi \), and \( \theta = 2\pi \).

The cosine curve, like the sine curve, also represents periodic phenomena. The key difference is its phase shift, which causes it to start at its peak value.
Periodicity
Periodicity is a crucial concept in understanding trigonometric functions like sine and cosine. Periodicity refers to the repeating nature of a function or wave that occurs at regular intervals. For sine and cosine, this regular interval is known as the period.
  • A full cycle of a sine or cosine function before repeating is called one period.
  • The standard mathematical period for sine and cosine functions in radians is \(2\pi\).
  • In degrees, the period corresponds to 360 degrees.

This inherent periodic nature makes these functions highly valuable in modeling cyclic phenomena. Whether it is electrical currents, sound, or even seasonal patterns, understanding periodicity helps us predict and analyze repetitive occurrences. When working with these functions, always consider the period as it dictates how often the pattern repeats. Remember, one complete cycle is called a period.

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Most popular questions from this chapter

Consider the functions \(f(x)=\sin x\) 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?

Write an algebraic expression that is equivalent to the given expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7.)\) \(\sin (\arccos x)\)

Use an inverse trigonometric function to write \(\theta\) as a function of \(x .\)

Find the exact value of the expression. (Hint: Sketch a right triangle.) \(\sin \left[\arccos \left(-\frac{2}{3}\right)\right]\)

Use a graphing utility to graph the function. \(f(x)=\pi-\sin ^{-1}\left(\frac{2}{3}\right)\)
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