4.2 Q-4

Question Statement

The $x$-$y$ coordinate axes are rotated about the origin through the indicated angle. The new axes are OX and OY. Find the XY-coordinates of the point $P$ with the given XY coordinates.

ii. $P(-5, 3), \theta = 30^{\circ}$

---

Background and Explanation

In problems involving the rotation of coordinate axes, we use rotation formulas to transform the coordinates of a point from the original $xy$ system to the new $XY$ system. The formulas are based on trigonometric identities, where the new coordinates $X$ and $Y$ are expressed in terms of the original coordinates $x$ and $y$ and the rotation angle $\theta$.

---

Solution

We are given the coordinates of point $P(-5, 3)$ in the original $xy$ system, and we need to find the new coordinates after rotating the axes by an angle $\theta = 30^\circ$.

Step 1: Apply the rotation formulas

The general rotation formulas are:

$$X = x \cos \theta + y \sin \theta$$ $$Y = -x \sin \theta + y \cos \theta$$

Substitute the given values of $x = -5$, $y = 3$, and $\theta = 30^\circ$:

For $X$:

$$X = (-5) \cos 30^\circ + 3 \sin 30^\circ$$

Using known values for $\cos 30^\circ = \frac{\sqrt{3}}{2}$ and $\sin 30^\circ = \frac{1}{2}$:

$$X = (-5) \left(\frac{\sqrt{3}}{2}\right) + 3 \left(\frac{1}{2}\right)$$ $$X = \frac{-5 \sqrt{3}}{2} + \frac{3}{2}$$ $$X = \frac{-5 \sqrt{3} + 3}{2}$$

For $Y$:

$$Y = -(-5) \sin 30^\circ + 3 \cos 30^\circ$$

Substitute the values for $\sin 30^\circ$ and $\cos 30^\circ$:

$$Y = 5 \left(\frac{1}{2}\right) + 3 \left(\frac{\sqrt{3}}{2}\right)$$ $$Y = \frac{5}{2} + \frac{3\sqrt{3}}{2}$$ $$Y = \frac{5 + 3\sqrt{3}}{2}$$

Thus, the new coordinates of the point $P$ in the rotated coordinate system are:

$$P\left(\frac{-5 \sqrt{3} + 3}{2}, \frac{5 + 3 \sqrt{3}}{2}\right)$$

---

Key Formulas or Methods Used

• Rotation Transformation Formula:
  • $X = x \cos \theta + y \sin \theta$
  • $Y = -x \sin \theta + y \cos \theta$

Where:

• $x, y$ are the coordinates in the original system
• $X, Y$ are the coordinates in the rotated system
• $\theta$ is the angle of rotation

---

Summary of Steps

1. Write down the rotation transformation formulas:

   • $X = x \cos \theta + y \sin \theta$
   • $Y = -x \sin \theta + y \cos \theta$

2. Substitute the given values for $x = -5$, $y = 3$, and $\theta = 30^\circ$ into the formulas.

3. Solve for $X$:

   • Use the trigonometric values for $\cos 30^\circ$ and $\sin 30^\circ$ to calculate $X$.

4. Solve for $Y$:

   • Use the same trigonometric values to calculate $Y$.

5. Write the final result for the new coordinates $P$ in the rotated system:

   • $P\left(\frac{-5 \sqrt{3} + 3}{2}, \frac{5 + 3 \sqrt{3}}{2}\right)$