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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![### Understanding Quadrants in Trigonometry
#### Problem Statement:
**Name the quadrant in which the angle θ lies.**
Given conditions:
\[ \sin \theta > 0, \quad \cos \theta > 0 \]
#### Question:
The angle θ lies in which quadrant?
**Options:**
- III
- IV
- I
- II
---
### Explanation of Solution:
In trigonometry, an angle θ in the coordinate system is located in one of four quadrants:
- **Quadrant I**: \( 0^\circ < \theta < 90^\circ \) (both \(\sin \theta\) and \(\cos \theta\) are positive)
- **Quadrant II**: \( 90^\circ < \theta < 180^\circ \) (\(\sin \theta\) is positive, \(\cos \theta\) is negative)
- **Quadrant III**: \( 180^\circ < \theta < 270^\circ \) (both \(\sin \theta\) and \(\cos \theta\) are negative)
- **Quadrant IV**: \( 270^\circ < \theta < 360^\circ \) (\(\sin \theta\) is negative, \(\cos \theta\) is positive)
Given the conditions:
\[ \sin \theta > 0 \]
\[ \cos \theta > 0 \]
Both sine and cosine functions are positive in **Quadrant I**.
Therefore, the angle θ lies in **Quadrant I**.
- Option I
Ensure to understand the properties of trigonometric functions and their signs in each quadrant when determining the location of an angle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e81cca1-46d7-47ec-8ce2-a27a4cdf5512%2F21e8a99d-82f6-4155-802a-c458f0bc5c07%2Fsu6zm7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Quadrants in Trigonometry
#### Problem Statement:
**Name the quadrant in which the angle θ lies.**
Given conditions:
\[ \sin \theta > 0, \quad \cos \theta > 0 \]
#### Question:
The angle θ lies in which quadrant?
**Options:**
- III
- IV
- I
- II
---
### Explanation of Solution:
In trigonometry, an angle θ in the coordinate system is located in one of four quadrants:
- **Quadrant I**: \( 0^\circ < \theta < 90^\circ \) (both \(\sin \theta\) and \(\cos \theta\) are positive)
- **Quadrant II**: \( 90^\circ < \theta < 180^\circ \) (\(\sin \theta\) is positive, \(\cos \theta\) is negative)
- **Quadrant III**: \( 180^\circ < \theta < 270^\circ \) (both \(\sin \theta\) and \(\cos \theta\) are negative)
- **Quadrant IV**: \( 270^\circ < \theta < 360^\circ \) (\(\sin \theta\) is negative, \(\cos \theta\) is positive)
Given the conditions:
\[ \sin \theta > 0 \]
\[ \cos \theta > 0 \]
Both sine and cosine functions are positive in **Quadrant I**.
Therefore, the angle θ lies in **Quadrant I**.
- Option I
Ensure to understand the properties of trigonometric functions and their signs in each quadrant when determining the location of an angle.
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