Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Please answer the question in the picture below. Show all work. Thank you
![**Question:**
What is the size of the acute angle formed by the x-axis and the line \(3x + 2y = 12\)?
**Explanation:**
To find the size of the acute angle formed by a line and the x-axis, we can use the line's slope. The equation of the line is in standard form. We need to convert it to slope-intercept form (\(y = mx + b\)) to identify the slope (\(m\)).
Starting with the given equation:
\[ 3x + 2y = 12 \]
Solve for \(y\):
\[ 2y = -3x + 12 \]
\[ y = -\frac{3}{2}x + 6 \]
The slope of the line (\(m\)) is \(-\frac{3}{2}\).
The angle \(\theta\) with the x-axis can be found using the formula:
\[ \tan(\theta) = |m| \]
\[ \tan(\theta) = \left| -\frac{3}{2} \right| = \frac{3}{2} \]
Thus, the acute angle \(\theta\) is:
\[ \theta = \tan^{-1}\left(\frac{3}{2}\right) \]
Using a calculator, find:
\[ \theta = \tan^{-1}\left(\frac{3}{2}\right) \approx 56.31^\circ \]
Therefore, the size of the acute angle is approximately \(56.31^\circ\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdead4581-e5fd-4d6e-8aad-fa19ced9c018%2F2a3ffd3c-aeb4-4528-8762-df86e345956c%2Fbndhfhl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
What is the size of the acute angle formed by the x-axis and the line \(3x + 2y = 12\)?
**Explanation:**
To find the size of the acute angle formed by a line and the x-axis, we can use the line's slope. The equation of the line is in standard form. We need to convert it to slope-intercept form (\(y = mx + b\)) to identify the slope (\(m\)).
Starting with the given equation:
\[ 3x + 2y = 12 \]
Solve for \(y\):
\[ 2y = -3x + 12 \]
\[ y = -\frac{3}{2}x + 6 \]
The slope of the line (\(m\)) is \(-\frac{3}{2}\).
The angle \(\theta\) with the x-axis can be found using the formula:
\[ \tan(\theta) = |m| \]
\[ \tan(\theta) = \left| -\frac{3}{2} \right| = \frac{3}{2} \]
Thus, the acute angle \(\theta\) is:
\[ \theta = \tan^{-1}\left(\frac{3}{2}\right) \]
Using a calculator, find:
\[ \theta = \tan^{-1}\left(\frac{3}{2}\right) \approx 56.31^\circ \]
Therefore, the size of the acute angle is approximately \(56.31^\circ\).
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