Problem 2 - Algebra and Trigonometry

Question

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(-2,1), \quad \phi=30^{\circ}$$

Step-by-Step Solution

Answer

The new coordinates are approximately \\( (X, Y) \approx (-1.232, 1.866) \\).

Step 1: Understanding Rotation

In this problem, we're given a point \((-2, 1)\)\ and asked to find its new coordinates \( (X, Y) \) after the coordinate axes are rotated by an angle \( heta = 30^{\circ} \). This requires use of rotation formulas.

Step 2: Using Rotation Formulas

The rotation transformation formulas are: \[ X = x \cos(\theta) + y \sin(\theta) \Y = -x \sin(\theta) + y \cos(\theta) \ \] where \( x = -2 \), \( y = 1 \) \ and \( \theta = 30^{\circ} \). We'll substitute these values into the formulas.

Step 3: Substitute Values for New X

Substituting into the formula for \( X \) gives: \[ X = (-2) \cos(30^{\circ}) + 1 \sin(30^{\circ}) \]\[ \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \, \text{and} \, \sin(30^{\circ}) = \frac{1}{2} \]\[ X = (-2) \left( \frac{\sqrt{3}}{2} \right) + 1 \left( \frac{1}{2} \right) = -\sqrt{3} + \frac{1}{2} \]

Step 4: Substitute Values for New Y

Substituting into the formula for \ Y \ gives: \[ Y = -(-2) \sin(30^{\circ}) + 1 \cos(30^{\circ}) \]\[ Y = 2 \left( \frac{1}{2} \right) + 1 \left( \frac{\sqrt{3}}{2} \right) = 1 + \frac{\sqrt{3}}{2} \]

Step 5: Compute Numeric Values

Let's compute the exact numeric values from the expressions: \[ X = -\sqrt{3} + \frac{1}{2} \approx -1.232 \] \[ Y = 1 + \frac{\sqrt{3}}{2} \approx 1.866 \]

Key Concepts

Rotation Formulas

Rotation formulas are essential in transforming coordinates due to a change in orientation of the axes. When the axes are rotated by an angle \( \theta \), the position of a point on the plane changes accordingly. These transformations can be succinctly captured using the formulas:

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

Here, \((x, y)\) are the original coordinates, while \((X, Y)\) are the coordinates after rotation. The angle \( \theta \) represents the degree of rotation in the counter-clockwise direction. These formulas are derived from the basic understanding of how a point's position shifts along the circular path in the xy-plane, maintaining its radius but changing its angular position. Understanding these formulas is key to manipulating and visualizing transformed coordinates in rotated axes.

Trigonometric Functions

Trigonometric functions, namely sine and cosine, play a crucial role in understanding coordinate rotation. They relate the angles of a triangle to the lengths of its sides, which are essential in rotation contexts.

• \( \cos(\theta) \) represents the adjacent-to-hypotenuse ratio on the right triangle formed by the axes and the point's vector.
• \( \sin(\theta) \) represents the opposite-to-hypotenuse ratio on the same triangle.

For example, - When \( \theta = 30^{\circ} \), it's well-known that \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \) and \( \sin(30^{\circ}) = \frac{1}{2} \).These values allow us to compute the exact changes in x and y coordinates under rotation. The sine and cosine functions serve as the bridge that helps the coordinates "rotate" while keeping the relative distances consistent. Hence, trigonometric identities ensure, mathematically, that the transformation preserves distances and angles.

Coordinate Axes Transformation

Coordinate axes transformation is a fundamental concept in geometry and linear algebra that allows us to describe a point's position relative to rotated axes. By rotating the axes rather than the point itself, the same point assumes a new position in the newly transformed coordinate system. This transformation is visually analogous to spinning a compass rose where north pivots to northeast, changing the definition of directions. Similarly, the geometric transformation relies heavily on rotation matrices, illustrated as:

• \( \begin{bmatrix} \cos(\theta) & \sin(\theta) \ -\sin(\theta) & \cos(\theta) \end{bmatrix} \)

This matrix method simplifies the computation and understanding of how each axis shifts accordingly. In practical terms, such transformations enable modeling of various physical systems in different orientations without physically rotating the object itself. Understanding this concept is crucial in fields ranging from robotics to aerospace, where rotations must be accounted for system modeling and navigation.