The axes are rotated through an angle of л/3 in the anticlockwise direction with respect to (0, 0). Find the coordinates of point (4, 2) (w.r.t. old coordinate system) in the new coordinates system.
The axes are rotated through an angle of л/3 in the anticlockwise direction with respect to (0, 0). Find the coordinates of point (4, 2) (w.r.t. old coordinate system) in the new coordinates system.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
The axes are rotated through an angle of \(\pi/3\) in the anticlockwise direction with respect to (0, 0). Find the coordinates of point (4, 2) (with respect to the old coordinate system) in the new coordinate system.
**Explanation:**
When the coordinate axes are rotated by an angle \(\theta\), the new coordinates \((x', y')\) of a point \((x, y)\) can be found using the following transformation formulas:
\[
x' = x \cos \theta + y \sin \theta
\]
\[
y' = -x \sin \theta + y \cos \theta
\]
For this problem:
- \(\theta = \pi/3\)
- Original coordinates: \((x, y) = (4, 2)\)
Substitute these values into the formulas to find the new coordinates.
1. Calculate \(x'\):
\[
x' = 4 \cos(\pi/3) + 2 \sin(\pi/3)
\]
2. Calculate \(y'\):
\[
y' = -4 \sin(\pi/3) + 2 \cos(\pi/3)
\]
**Key Concepts:**
- **Rotation of Axes**: When axes are rotated, the position of a point changes relative to the new axes but remains the same in absolute space.
- **Trigonometric Identities**: Knowledge of \(\cos\) and \(\sin\) values for key angles like \(\pi/3\) (\(\cos(\pi/3)=1/2\), \(\sin(\pi/3)=\sqrt{3}/2\)) are used to simplify calculations.
This approach is crucial in various applications such as computer graphics, physics, and engineering where coordinate transformation is necessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4f4865a-ef7d-48b6-9e5b-0e85084b2d7c%2Fb7c05ddb-1521-4df9-9606-09c2a468b59d%2Fmrytwl3j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The axes are rotated through an angle of \(\pi/3\) in the anticlockwise direction with respect to (0, 0). Find the coordinates of point (4, 2) (with respect to the old coordinate system) in the new coordinate system.
**Explanation:**
When the coordinate axes are rotated by an angle \(\theta\), the new coordinates \((x', y')\) of a point \((x, y)\) can be found using the following transformation formulas:
\[
x' = x \cos \theta + y \sin \theta
\]
\[
y' = -x \sin \theta + y \cos \theta
\]
For this problem:
- \(\theta = \pi/3\)
- Original coordinates: \((x, y) = (4, 2)\)
Substitute these values into the formulas to find the new coordinates.
1. Calculate \(x'\):
\[
x' = 4 \cos(\pi/3) + 2 \sin(\pi/3)
\]
2. Calculate \(y'\):
\[
y' = -4 \sin(\pi/3) + 2 \cos(\pi/3)
\]
**Key Concepts:**
- **Rotation of Axes**: When axes are rotated, the position of a point changes relative to the new axes but remains the same in absolute space.
- **Trigonometric Identities**: Knowledge of \(\cos\) and \(\sin\) values for key angles like \(\pi/3\) (\(\cos(\pi/3)=1/2\), \(\sin(\pi/3)=\sqrt{3}/2\)) are used to simplify calculations.
This approach is crucial in various applications such as computer graphics, physics, and engineering where coordinate transformation is necessary.
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