/*! 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 17 a) Arrange the following values ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 17

a) Arrange the following values of sine in increasing order. \(\sin 1, \sin 2, \sin 3, \sin 4\) b) Show what the four values represent on a diagram of the unit circle. Use your diagram to justify the order from part a). c) Predict the correct increasing order for \(\cos 1, \cos 2, \cos 3,\) and \(\cos 4 .\) Check with a calculator. Was your prediction correct?

Short Answer

Expert verified
The increasing order of sine values is \( \text{sin 4}, \text{sin 3}, \text{sin 1}, \text{sin 2} \). For cosine values, the increasing order is \( \text{cos 3}, \text{cos 4}, \text{cos 2}, \text{cos 1} \).

Step by step solution

01

- Find the values of sine

Using a calculator, find the values for \(\text{sin 1}, \text{sin 2}, \text{sin 3}, \text{sin 4}\): - \(\text{sin 1} \approx 0.8415\)- \(\text{sin 2} \approx 0.9093\)- \(\text{sin 3} \approx 0.1411\)- \(\text{sin 4} \approx -0.7568\)
02

- Arrange the values in increasing order

Order the sine values from lowest to highest: - \(\text{sin 4} \approx -0.7568\)- \(\text{sin 3} \approx 0.1411\)- \(\text{sin 1} \approx 0.8415\)- \(\text{sin 2} \approx 0.9093\)
03

- Plot the values on the unit circle

Draw a unit circle and mark the angles \(\theta = 1\), \(\theta = 2\), \(\theta = 3\) and \(\theta = 4\) (in radians). Observe the sine values as the y-coordinates of these points.
04

- Justify the order using the diagram

From the unit circle diagram, observe that as you move from \(\theta = 4\) to \(\theta = 3\), \(\theta = 1\), and \(\theta = 2\), you see the y-coordinates (sine values): \(\text{sin 4} \approx -0.7568\), \(\text{sin 3} \approx 0.1411\), \(\text{sin 1} \approx 0.8415\), \(\text{sin 2} \approx 0.9093\). This confirms the order.
05

- Predict the order for cosine values

From the unit circle properties, cosine values are the x-coordinates. Predict the order for \(\text{cos 1}, \text{cos 2}, \text{cos 3}, \text{cos 4}\). Typically, these values decrease as the angle moves counterclockwise.
06

- Verify with calculator

Using a calculator, find the values for \(\text{cos 1}, \text{cos 2}, \text{cos 3}, \text{cos 4}\): - \(\text{cos 1} \approx 0.5403\)- \(\text{cos 2} \approx -0.4161\)- \(\text{cos 3} \approx -0.989992\)- \(\text{cos 4} \approx -0.6536\). The increasing order is \(\text{cos 3} \approx -0.989992\), \(\text{cos 4} \approx -0.6536\), \(\text{cos 2} \approx -0.4161\), \(\text{cos 1} \approx 0.5403\).

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.

Understanding the Sine Function
The sine function is a fundamental concept in trigonometry. It maps an angle, measured in radians, to a value between -1 and 1 on the unit circle. This value represents the y-coordinate of the point on the unit circle at that angle.

Key characteristics of the sine function include:
  • The sine of 0 is 0
  • The sine of \(\frac{\beta}{2}\) is 1
  • The sine of \(\beta\) is 0
  • The sine of \(\frac{3\beta}{2}\) is -1
On the unit circle, the sine values are highest at \(\frac{\beta}{2}\) and lowest at \(\frac{3\beta}{2}\).

In our problem, we have angles such as \(1, 2, 3\), and \(4\) radians which all land us in different quadrants.
  • For \(1\) radian, the sine value is approximately 0.8415
  • For \(2\) radians, the sine value is approximately 0.9093
  • For \(3\) radians, the sine value is approximately 0.1411
  • For \(4\) radians, the sine value is approximately -0.7568
This helps us arrange the values from lowest to highest as: -0.7568, 0.1411, 0.8415, and 0.9093.

Using the unit circle, it's clear that the y-coordinates of these angles justify our order.
Exploring the Cosine Function
The cosine function is another key trigonometric function. It maps an angle, measured in radians, to a value between -1 and 1 on the unit circle. This value represents the x-coordinate of the point on the unit circle at that angle.

Key characteristics of the cosine function include:
  • The cosine of 0 is 1
  • The cosine of \(\frac{\beta}{2}\) is 0
  • The cosine of \(\beta\) is -1
  • The cosine of \(\frac{3\beta}{2}\) is 0
On the unit circle, the cosine values are maximum at 0 radians and minimum at \(\beta\) radians.

In our problem, the angles \(1, 2, 3,\) and \(4\) radians yield the following cosine values:
  • For \(1\) radian, the cosine value is approximately 0.5403
  • For \(2\) radians, the cosine value is approximately -0.4161
  • For \(3\) radians, the cosine value is approximately -0.989992
  • For \(4\) radians, the cosine value is approximately -0.6536
When we arrange these values from lowest to highest, we get: -0.989992, -0.6536, -0.4161, and 0.5403.

This ordering aligns with the general behavior of cosine values as the angle moves counterclockwise.
Understanding Radians
Radians are a way of measuring angles. They provide a direct relationship between the angle and the arc length on a unit circle.

One radian is the angle at the center of a circle that subtends an arc length equal to the radius of the circle. There are approximately \(2\beta\) radians in a full circle. Hence, \(1, 2, 3,\), and \(4\) radians refer to angles of varying magnitudes around the circle.
  • \(1\) radian is just slightly more than \(57.3\) degrees.
  • \(2\) radians is slightly less than 120 degrees.
  • \(3\) radians is just under 180 degrees.
  • \(4\) radians is over 230 degrees.
Using radians to measure angles is essential in trigonometry as it simplifies many calculations and formulas.

To effectively use the unit circle for solving problems, it's crucial to understand how radians work and how they relate to the circle's coordinates. For instance, as you move counterclockwise from zero radians, both sine and cosine values start at their respective maxima (1 for cosine and 0 for sine) and cycle through their ranges.

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

Nora is required to solve the following trigonometric equation. \(9 \sin ^{2} \theta+12 \sin \theta+4=0, \theta \in\left[0^{\circ}, 360^{\circ}\right)\) Nora did the work shown below. Examine her work carefully. Identify any errors. Rewrite the solution, making any changes necessary for it to be correct. \(9 \sin ^{2} \theta+12 \sin \theta+4=0\) \((3 \sin \theta+2)^{2}=0\) $$ 3 \sin \theta+2=0 $$ Therefore. \(\sin \theta=-\frac{2}{3}\) Use a colculator. \(\sin ^{-1}\left(-\frac{2}{3}\right)=-41.8103149\) So, the reference ongle is 41.8 , to the neorest tenth of a degree Sine is negotive in quodrants II ond III. The solution in quadront II is \(180^{\circ}-41.8^{\circ}=138.2\) The solution in quadrant III is \(180^{\circ}+41.8=221.8\) Therefore, \(\theta=138.2^{\circ}\) ond \(\theta=221.8\), to the neorest tenth of a degree.

Determine the exact values of the other five trigonometric ratios under the given conditions. a) \(\sin \theta=\frac{3}{5}, \frac{\pi}{2}<\theta<\pi\) b) \(\cos \theta=\frac{-2 \sqrt{2}}{3},-\pi \leq \theta \leq \frac{3 \pi}{2}\) c) \(\tan \theta=\frac{2}{3},-360^{\circ}<\theta<180^{\circ}\) d) \(\sec \theta=\frac{4 \sqrt{3}}{3},-180^{\circ} \leq \theta \leq 180^{\circ}\)

Todd believes that \(180^{\circ}\) and \(270^{\circ}\) are solutions to the equation \(5 \cos ^{2} \theta=-4 \cos \theta .\) Show how you would check to determine whether Todd's solutions are correct.

Alice Through the Looking Glass by Lewis Carroll introduced strange new worlds where time ran backwards. Your challenge is to imagine a unit circle in which a positive rotation is defined to be clockwise. Assume the coordinate system remains as we know it. a) Draw a unit circle in which positive angles are measured clockwise from \((0,1) .\) Label where \(R\left(\frac{\pi}{6}\right), R\left(\frac{5 \pi}{6}\right), R\left(\frac{7 \pi}{6}\right)\) and \(\mathrm{R}\left(\frac{11 \pi}{6}\right)\) are on your new unit circle. b) What are the coordinates for the new \(\mathrm{R}\left(\frac{\pi}{6}\right)\) and \(\mathrm{R}\left(\frac{5 \pi}{6}\right) ?\) c) How do angles in this new system relate to conventional angles in standard position? d) How does your new system of angle measure relate to bearings in navigation? Explain.

For each point, sketch two coterminal angles in standard position whose terminal arm contains the point. Give one positive and one negative angle, in radians, where neither angle exceeds one full rotation. a) (3,5) b) (-2,-1) c) (-3,2) d) (5,-2)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

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.