/*! 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 89 Without drawing a graph, describ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 89

Without drawing a graph, describe the behavior of the basic sine curve.

Short Answer

Expert verified
The basic sine curve is a smooth, repetitive oscillation starting from the origin (0,0). It has a wavelength of \(2\pi\), meaning it repeats every \(2\pi\) units and an amplitude of 1, meaning it reaches a maximum height of 1 and a minimum height of -1. The curve crosses the x-axis at \(x = n\pi\) for all integers n.

Step by step solution

01

Define the Sine Function

The sine function, often denoted as sin(x), is a periodic function that describes a smooth, repetitive oscillation. It鈥檚 one of the basic functions in trigonometry. The standard sine curve y = sin(x) has a wavelength (the length of one whole wave) of \(2\pi\), an amplitude (the height from the middle to the top or bottom) of 1.
02

Explain the Key Points on the Sine Curve

The sine curve begins at the origin (0,0). As x increases, it goes up to a maximum height of 1 at \(x = \frac{\pi}{2}\), then comes down again and reaches the minimum height of -1 at \(x = \frac{3\pi}{2}\) before returning to 0 at \(x = 2\pi\). This pattern then repeats itself for all larger x-values.
03

Describe the Periodicity of the Sine Curve

The sine curve is periodic, meaning it repeats its pattern every \(2\pi\) units. This is called the period of the sine function. For all integer values of n, the curve has x-intercepts (where the curve crosses the x-axis) at \(x = n\pi\). Moreover, for the sine function, the maximum amplitude is 1, and the minimum amplitude is -1.

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!

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

Stonehenge, the famous "stone circle" in England, was built between 2750 B.C. and 1300 B.C. using solid stone blocks weighing over 99,000 pounds each. It required 550 people to pull a single stone up a ramp inclined at a \(9^{\circ}\) angle. Describe how right triangle trigonometry can be used to determine the distance the 550 workers had to drag a stone in order to raise it to a height of 30 feet. (IMAGE CANNOT COPY)

The number of hours of daylight, \(H,\) on day \(t\) of any given year (on January \(1, t=1\) ) in San Diego, California, can be modeled by the function $$H(t)=12+2.4 \sin \left[\frac{2 \pi}{365}(t-80)\right]$$ a. March \(21,\) the 80 th day of the year, is the spring equinox. Find the number of hours of daylight in San Diego on this day. b. June \(21,\) the 172 nd day of the year, is the summer solstice, the day with the maximum number of hours of daylight. Find, to the nearest tenth of an hour, the number of hours of daylight in San Diego on this day. c. December 21 , the 355 th day of the year, is the winter solstice, the day with the minimum number of hours of daylight. To the nearest tenth of an hour, find the number of hours of daylight in San Diego on this day.

How do we measure the distance between two points, \(A\) and \(B,\) on Earth? We measure along a circle with a center, \(C,\)at the center of Earth. The radius of the circle is equal to the distance from C to the surface. Use the fact that Earth is a sphere of radius equal to approximately 4000 miles to solve. If two points, \(A\) and \(B\), are 8000 miles apart, express angle \(\theta\) in radians and in degrees.

In Exercises \(9-16\), evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$\csc \pi$$

Use a calculator to find the value of each expression rounded to two decimal places. $$\sin ^{-1}(-0.32)$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.