Problem 37 - Algebra and Trigonometry

Question

Do you expect that the distance between two points is invariant under rotation? Prove your answer by comparing the distance \(d(P, Q)\) and \(d\left(P^{\prime}, Q^{\prime}\right)\) where \(P^{\prime}\) and \(Q^{\prime}\) are the images of \(P\) and \(Q\) under a rotation of axes.

Step-by-Step Solution

Answer

Yes, the distance is invariant under rotation, proved using trigonometric identities and the distance formula.

Step 1: Understand the Problem

The problem asks if the distance between two points remains the same after a rotation. We need to show mathematically that the distance between points doesn't change when the coordinate system is rotated.

Step 2: Define the Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:\[d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Step 3: Apply Rotation to Points

Under a rotation by an angle \(\theta\), the coordinates of point \((x, y)\) transform to \((x', y')\) such that:\[x' = x \cos \theta - y \sin \theta\]\[ y' = x \sin \theta + y \cos \theta\]

Step 4: Calculate Transformed Distances

Now, calculate the distance between the transformed points \(P'\) and \(Q'\). Let \((x_1', y_1')\) and \((x_2', y_2')\) be the rotated coordinates of \(P\) and \(Q\), respectively:\[d(P', Q') = \sqrt{(x_2' - x_1')^2 + (y_2' - y_1')^2}\]

Step 5: Substitute the Transformed Coordinates

Substituting the formulas for \(x'\) and \(y'\) from Step 3 into the distance formula from Step 4:\[(x_2' - x_1') = (x_2 \cos \theta - y_2 \sin \theta) - (x_1 \cos \theta - y_1 \sin \theta)\]\[(y_2' - y_1') = (x_2 \sin \theta + y_2 \cos \theta) - (x_1 \sin \theta + y_1 \cos \theta)\]

Step 6: Simplify the Expression

Simplifying:\[d(P', Q') = \sqrt{((x_2-x_1) \cos \theta - (y_2 - y_1) \sin \theta)^2 + ((x_2-x_1) \sin \theta + (y_2-y_1) \cos \theta)^2}\]

Step 7: Use Trigonometric Identity

Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\), the expression inside the square root simplifies to:\[ (x_2-x_1)^2 + (y_2-y_1)^2 \]This confirms that:\[d(P', Q') = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = d(P, Q)\]

Step 8: Conclusion

We have shown that the distance between the two points remains unchanged after the rotation of the axes. Hence, the distance between two points is invariant under rotation.

Key Concepts

rotation of axes

The concept of rotating the axes is fundamental in understanding how shapes and distances are preserved, even when the coordinate system changes. This transformation involves rotating the entire Cartesian plane around the origin by a specified angle \(\theta\). During this rotation:

• The positions of points on the plane change.
• Coordinates transform using trigonometric functions.

It's crucial because despite these coordinate changes, certain properties, like distances, remain unaffected. For instance, if a point \((x, y)\) is rotated, its new coordinates \((x', y')\) are determined by:\[x' = x \cos \theta - y \sin \theta\]\[y' = x \sin \theta + y \cos \theta\]These formulas make use of the fundamental trigonometric nature of rotation, ensuring that the geometry of the shape, like its size and orientation relative to other properties, is preserved.

distance formula

The distance formula is a key tool in geometry, helping us find the distance between two points in the Cartesian plane. It is derived from the Pythagorean theorem and measured as:\[d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] This formula relies on the notion that a straight line between two points forms the hypotenuse of a right triangle, with the horizontal and vertical differences between points constituting the other two sides.
What's remarkable about the distance formula is its simplicity and versatility for use in various mathematics fields.

• Helps in analyzing point positions.
• Ensures quantities like length remain invariant under transformations.

It is the backbone of proving that distances don't change even when coordinates do.

coordinate transformation

Coordinate transformation is where the coordinates of a point change when the axes are moved, reflecting the same geometric entity in a different orientation.This involves a mathematical renovation of the coordinate system, often through rotation or translation.
The transformations can be represented as equations, which in a rotation scenario, redefine the coordinates of a point \((x, y)\) as:

• \(x' = x \cos \theta - y \sin \theta\)
• \(y' = x \sin \theta + y \cos \theta\)

Although this might appear as altering coordinates, it essentially maintains:

• The geometric properties of points and lines.
• Ensures that distances and angles remain the same.

Coordinate transformations help to solve complex problems by simplifying them to a more manageable frame of reference.

trigonometric identities

Trigonometric identities play an essential role in simplifying expressions during the analysis of rotation impacts on the distance between two points.
In the context of axis rotation, these identities help derive simpler mathematical expressions. A critical identity is:\[\cos^2 \theta + \sin^2 \theta = 1\]This identity allows us to reconcile the components of the transformed distance formula by demonstrating that the sum of the squares of the sine and cosine components equals one.
By using this identity, we can simplify the distance formula after rotation:\[d(P', Q') = \sqrt{((x_2-x_1) \cos \theta - (y_2 - y_1) \sin \theta)^2 + ((x_2-x_1) \sin \theta + (y_2-y_1) \cos \theta)^2}\]Ultimately, it shows that the value hasn't changed from the original distance:\[d(P, Q) = d(P', Q')\] This confirms that distances remain invariant, proving the robustness of trigonometric identities in rotation contexts.