/*! 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 94 What is a periodic function? Why... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 94

What is a periodic function? Why are the sine and cosine functions periodic?

Short Answer

Expert verified
A periodic function is one that repeats itself after a regular interval, known as the period. The sine and cosine functions are periodic because their values repeat after an interval of \(2\pi\), representing a full rotation around a unit circle.

Step by step solution

01

Understanding Periodic Functions

A function is considered 'periodic' if it repeats its values at regular intervals or 'periods'. This simply means that for some number \( p \), if you add \( p \) to the input of the function, the output will be the same. So if a function \( f \) is periodic with period \( p \), then for all \( x \) in its domain, \( f(x + p) = f(x) \).
02

Defining Sine and Cosine Functions

The sine and cosine functions are two basic functions in mathematics, particularly in trigonometry. They are defined for real numbers and usually expressed with the help of the unit circle (a circle with radius 1). Here sine gives the y-coordinate (up and down), and cosine gives the x-coordinate (left and right). They are often used to model periodic phenomena.
03

Explaining Periodicity of Sine and Cosine

Both the sine and cosine functions are periodic. They have a period of \(2\pi\), which means that sine and cosine 'repeat' every \(2\pi\) units. Mathematically, this means for all \(x\), we have (\(\sin(x + 2\pi) = \sin(x)\)) and (\(\cos(x + 2\pi) = \cos(x)\)). This behavior is explained by the fact that adding \(2\pi\) to the argument of sine or cosine means going one full circle around the unit circle, which is again the starting point.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Sine and Cosine Functions
When we talk about sine and cosine functions, we鈥檙e diving into the heart of trigonometry. Imagine standing at the center of a clock, with hands moving around. The sine function tells us how high the hand is (the y-coordinate) and the cosine function tells us how far to the side it is (the x-coordinate).

But here鈥檚 the thing: no matter how far the hands move, they always loop back to the same spot every 12 hours. It's this looping, this repeating, which makes these functions periodic. They鈥檙e like your favorite song's chorus, coming back over and over again. And just like a song has a specific length, sine and cosine repeat every \(2\pi\) units, which in terms of our clock, is like a full day鈥檚 rotation.
Trigonometry
Now, let鈥檚 break down trigonometry. Trig is all about the study of triangles, particularly right triangles, and the relationships between their angles and sides. Like a universal language, it helps us understand patterns and solve problems involving triangles and circles.

Picture slicing a pizza into perfect triangles 鈥� the angles and lengths of the crust (hypotenuse) versus the cheesy sides (the two legs) always have a unique relationship. Sine and cosine are just ratios from these slices. So, in trigonometry, they're like the secret sauce to solving many geometrical and real-world conundrums, from building bridges to sending satellites into orbit.
Unit Circle
Speaking of pizza, a unit circle is sort of like the perfect personal-sized pizza, with a radius of just one bite 鈥� I mean, unit. It's a super useful tool in trigonometry because every point on the circle鈥檚 edge can represent the endpoint of the hypotenuse of a right-angled triangle, where the other two sides are on the 鈥榵鈥� and 鈥榶鈥� axis.

As you go round this mini pizza, you can sample every possible value for sine and cosine, because they correspond to the 'y' and 'x' coordinates at each point. Just remember, once you make a full circle, all the flavors 鈥� or numerical values 鈥� start repeating. This is why the unit circle is a powerhouse when it comes to understanding angles and their corresponding sine and cosine values.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Exercises \(117-119\) will help you prepare for the material covered in the next section. In each exercise, complete the table of coordinates. Do not use a calculator. $$y=3 \sin \frac{\pi}{2} x$$ $$\begin{array}{|l|l|l|l|l|l|l|l|l|} \hline x & 0 & \frac{1}{3} & 1 & \frac{5}{3} & 2 & \frac{7}{3} & 3 & \frac{11}{3} & 4 \\ \hline y & & & & & & & \\ \hline \end{array}$$ After completing this table of coordinates, plot the nine ordered pairs as points in a rectangular coordinate system. Then connect the points with a smooth curve.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When an angle's measure is given in terms of \(\pi,\) I know that it's measured using radians.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I convert degrees to radians, I multiply by \(1,\) choosing \(\frac{\pi}{180^{\circ}}\) for 1

$$\text { Prove that if } x>0, \tan ^{-1} x+\tan ^{-1} \frac{1}{x}=\frac{\pi}{2}$$

You invested \(\$ 3000\) in two accounts paying \(6 \%\) and \(8 \%\) annual interest. If the total interest earned for the year was \(\$ 230,\) how much was invested at each rate? (Section P.8, Example 5).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.