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Given the following function, choose the graph with the two possible angles with the domain 0 sA S 360º. sine = -

Given the following function, choose the graph with the two possible angles with the domain 0 sA S 360º. sine = -

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Transcription and Explanation for Educational Website** **Problem Statement:** Given the following function, choose the graph with the two possible angles with the domain \(0 \leq \theta \leq 360^\circ\). \[ \sin \theta = -\frac{2}{3} \] **Graph Explanation:** The graph is a coordinate plane with x and y axes, each marked with arrows pointing in positive directions. It shows two angles, \(\theta\) and \(\theta_1\), in different colors. - **Axes and Quadrants:** The graph is divided into four quadrants by the x-axis and y-axis, which intersect at the origin (0,0). - **Angles:** - The angle \(\theta\) is represented by a blue arrow, rotating clockwise from the positive y-axis. - The angle \(\theta_1\) is shown as a red arrow, also rotating clockwise but from a different origin angle. - **Sine Function Representation:** The negative sine value suggests that both angles are in quadrants where sine is negative, namely the third and fourth quadrants. This graph visually represents the possible angles \(\theta\) formed by the given sine function within the specified range.
Transcribed Image Text:**Transcription and Explanation for Educational Website** **Problem Statement:** Given the following function, choose the graph with the two possible angles with the domain \(0 \leq \theta \leq 360^\circ\). \[ \sin \theta = -\frac{2}{3} \] **Graph Explanation:** The graph is a coordinate plane with x and y axes, each marked with arrows pointing in positive directions. It shows two angles, \(\theta\) and \(\theta_1\), in different colors. - **Axes and Quadrants:** The graph is divided into four quadrants by the x-axis and y-axis, which intersect at the origin (0,0). - **Angles:** - The angle \(\theta\) is represented by a blue arrow, rotating clockwise from the positive y-axis. - The angle \(\theta_1\) is shown as a red arrow, also rotating clockwise but from a different origin angle. - **Sine Function Representation:** The negative sine value suggests that both angles are in quadrants where sine is negative, namely the third and fourth quadrants. This graph visually represents the possible angles \(\theta\) formed by the given sine function within the specified range.
The image contains six graphs, each illustrating the concept of rotation of vectors on a coordinate plane. **Graph 1:** - A black vector is positioned on the grid, making an angle \(\theta_1\) with the positive x-axis. - A circular arc highlights the angle \(\theta_1\) in red and a complementary angle \(\theta\) in blue, made by the vector with the negative y-axis. **Graph 2:** - Two vectors are shown: a red vector making an angle \(\theta_1\) with the positive x-axis, and a blue vector making an angle \(\theta\) with the same axis. - The rotation from the red to the blue vector is depicted by a circular arc along the angle \(\theta\). **Graph 3:** - The red vector is shown with the blue vector rotated to a new position. - The diagram focuses on \(\theta_1\) formed by the red vector and a new blue vector position oriented at \(\theta - \theta_1\). **Graph 4:** - A red vector makes an angle \(\theta_1\) with the positive x-axis, while a blue vector lies rotated around to make angle \(\theta\) with the negative x-axis. - An arc marks the rotation from the blue vector to the position of the red vector. **Graph 5:** - The blue vector is in a different orientation, making angles \(\theta_1\) with the negative x-axis. - The arcs illustrate the relationship between these angles and the new positions of the vectors. **Graph 6:** - This graph looks similar but focuses primarily on the blue vector making an angle \(\theta\) with the positive x-axis. - There is an absence of the secondary vector. These graphs visually represent vector transformations, rotations, and the relationships between different angular orientations on a Cartesian plane.
Transcribed Image Text:The image contains six graphs, each illustrating the concept of rotation of vectors on a coordinate plane. **Graph 1:** - A black vector is positioned on the grid, making an angle \(\theta_1\) with the positive x-axis. - A circular arc highlights the angle \(\theta_1\) in red and a complementary angle \(\theta\) in blue, made by the vector with the negative y-axis. **Graph 2:** - Two vectors are shown: a red vector making an angle \(\theta_1\) with the positive x-axis, and a blue vector making an angle \(\theta\) with the same axis. - The rotation from the red to the blue vector is depicted by a circular arc along the angle \(\theta\). **Graph 3:** - The red vector is shown with the blue vector rotated to a new position. - The diagram focuses on \(\theta_1\) formed by the red vector and a new blue vector position oriented at \(\theta - \theta_1\). **Graph 4:** - A red vector makes an angle \(\theta_1\) with the positive x-axis, while a blue vector lies rotated around to make angle \(\theta\) with the negative x-axis. - An arc marks the rotation from the blue vector to the position of the red vector. **Graph 5:** - The blue vector is in a different orientation, making angles \(\theta_1\) with the negative x-axis. - The arcs illustrate the relationship between these angles and the new positions of the vectors. **Graph 6:** - This graph looks similar but focuses primarily on the blue vector making an angle \(\theta\) with the positive x-axis. - There is an absence of the secondary vector. These graphs visually represent vector transformations, rotations, and the relationships between different angular orientations on a Cartesian plane.
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