Problem 5 - Algebra and Trigonometry

Question

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (3,-\sqrt{3}), \quad \phi=60^{\circ} $$

Step-by-Step Solution

Answer

The coordinates after rotation are \((0, -2\sqrt{3})\).

Step 1: Rotation Formula Identification

To determine the new coordinates after rotation, we need to use the rotation of axes formula. The new coordinates \((X, Y)\) after rotation by an angle \( \phi \) are given by:\[ X = x \cos \phi + y \sin \phi \]\[ Y = -x \sin \phi + y \cos \phi \] where \((x, y)\) are the original coordinates.

Step 2: Substituting Given Values

Substitute the given values into the rotation formulas. Here, \((x, y) = (3, -\sqrt{3})\) and \( \phi = 60^{\circ} \). Using the trigonometric values:\( \cos 60^{\circ} = \frac{1}{2} \) and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), substitute into the equations:\[ X = 3 \cdot \frac{1}{2} + (-\sqrt{3}) \cdot \frac{\sqrt{3}}{2} \]\[ Y = -3 \cdot \frac{\sqrt{3}}{2} + (-\sqrt{3}) \cdot \frac{1}{2} \]

Step 3: Calculating New X-coordinate

Calculate the new X-coordinate using the substitution:\[ X = 3 \cdot \frac{1}{2} + (-\sqrt{3}) \cdot \frac{\sqrt{3}}{2} = \frac{3}{2} - \frac{3}{2} = 0 \]

Step 4: Calculating New Y-coordinate

Calculate the new Y-coordinate using the substitution:\[ Y = -3 \cdot \frac{\sqrt{3}}{2} + (-\sqrt{3}) \cdot \frac{1}{2} = -\frac{3\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = -\frac{4\sqrt{3}}{2} = -2\sqrt{3} \]

Step 5: Conclusion of New Coordinates

After performing the calculations, we find that the point \((3, -\sqrt{3})\) rotates to \((0, -2\sqrt{3})\) when the axes are rotated by an angle of \(60^{\circ}\).

Key Concepts

Trigonometric Values

Understanding trigonometric values is crucial in calculating coordinate transformations during axis rotations. In trigonometry, angles are usually expressed in degrees or radians, and their corresponding sine and cosine values help determine the new rotated coordinates.
For example, at an angle of 60 degrees, the trigonometric values are well-established:

• The cosine of 60 degrees, written as \( \cos 60^{\circ} \), equals \( \frac{1}{2} \).
• The sine of 60 degrees, written as \( \sin 60^{\circ} \), equals \( \frac{\sqrt{3}}{2} \).

These values are derived from the properties of a 30-60-90 triangle, one of the special triangles in trigonometry.
By using these specific sine and cosine values, we can correctly rotate coordinates in the plane.

Coordinate Transformation

Coordinate transformation is a geometric process to change from one coordinate system to another. A common transformation involves rotating axes and obtaining new coordinates, typically denoted as \((X, Y)\).
The concept primarily involves rotation where the original coordinates \((x, y)\) get transformed when the axes are rotated by an angle \( \phi \). Using the rotation of axes formulas, we can compute these new coordinates based on the original inputs and the rotation angle.
In our example, the original coordinates \((3, -\sqrt{3})\) are transformed by rotating the axes through 60 degrees. This transformation recalculates the position of points while keeping the structure with respect to the new axis orientation.

Rotation Formula

The rotation formula provides a systematic way to derive the new coordinates of a point after a rotation around the origin by a given angle. It utilizes the effects of trigonometric ratios on the point’s position.
Here's the rotation formula for a point \((x, y)\) rotated by an angle \( \phi \):

• The new X-coordinate \( X = x \cos \phi + y \sin \phi \).
• The new Y-coordinate \( Y = -x \sin \phi + y \cos \phi \).

This formula adjusts the coordinates by factoring in the angle’s sine and cosine values.
In practical terms, it means for our example with a point \((3, -\sqrt{3})\) and angle 60 degrees:

• Compute \( X = 3 \cdot \frac{1}{2} + (-\sqrt{3}) \cdot \frac{\sqrt{3}}{2} = 0 \).
• Compute \( Y = -3 \cdot \frac{\sqrt{3}}{2} + (-\sqrt{3}) \cdot \frac{1}{2} = -2\sqrt{3} \).

The resulting coordinates \( (0, -2\sqrt{3}) \) clearly indicate how the rotation affects the original point's position.