02_Ex 4.2

Exercise Questions

Questions	Links
Q1. The two points P and $\mathrm{O}^{\prime}$ are given…	4.2 Q-1
Q2. The $x y$ - coordinate axes are translated…	4.2 Q-2
Q3. The $x y$-coordinates axes are rotated…	4.2 Q-3
Q4. The $x y$-coordinates axes are rotated…	4.2 Q-4

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Overview

In this exercise, we explore the concept of coordinate transformation, including rotation and translation of points in the Cartesian plane. These transformations are widely used in fields such as physics, engineering, computer graphics, and robotics for changing perspectives or shifting frames of reference.

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Key Concepts

• Coordinate Rotation: A process where the coordinate axes are rotated by an angle $\theta$ around the origin. The new coordinates of a point are determined using trigonometric functions of the angle.
• Trigonometric Functions:
  • Sine: $\sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}}$.
  • Cosine: $\cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}$.
  • Tangent: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.
• Rotation Formula: For a point $(x, y)$ rotated through an angle $\theta$, the new coordinates $(x', y')$ are given by:

$$\begin{aligned} x' &= x \cos(\theta) - y \sin(\theta), y' &= x \sin(\theta) + y \cos(\theta). \end{aligned}$$

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Important Formulas

1. Rotation of Coordinates: For a point $P(x, y)$ and an angle $\theta$:

$$\begin{aligned} x' &= x \cos(\theta) - y \sin(\theta), y' &= x \sin(\theta) + y \cos(\theta). \end{aligned}$$

2. Known Trigonometric Values:

   • $\sin(30^\circ) = \frac{1}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$.
   • $\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$.
   • $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, $\cos(60^\circ) = \frac{1}{2}$.

3. Distance Formula: To find the distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$

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Tips and Tricks

1. Understanding Angles: Always ensure that the angle $\theta$ is in the correct quadrant when applying the trigonometric identities. The signs of $\sin(\theta)$ and $\cos(\theta)$ change depending on the quadrant.
2. Check Coordinates: After performing the rotation, double-check that the new coordinates align with your expectations based on the angle of rotation.
3. Use Known Trigonometric Values: Memorize common angles like $30^\circ$, $45^\circ$, and $60^\circ$ and their sine and cosine values to simplify calculations.
4. Coordinate Significance: Be aware of the signs of the new coordinates, especially when rotating points that cross axes.

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Summary

In this exercise, we covered:

• Coordinate rotation to determine new coordinates of a point after rotation by an angle $\theta$.
• Translation of points with respect to new origin coordinates.
• Practical usage of trigonometric functions to simplify the process of transformation.

The key takeaway is that these transformations are fundamental in adjusting coordinate systems and are heavily used in fields that require spatial manipulation.

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Reference

By Sir Shahzad Sair:

By Great Science Academy: