Problem 6 - Precalculus: Mathematical for Calculus

Question

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(\sqrt{2}, 4 \sqrt{2}), \quad \phi=45^{\circ}$$

Step-by-Step Solution

Answer

The rotated coordinates are (5, 3).

Step 1: Understand the Problem

We are given a point in Cartesian coordinates, \(\sqrt{2}, 4 \sqrt{2}\), and a rotation angle, \phi = 45^{\circ}\. The task is to determine the new coordinates of the point after rotating the coordinate axes by this angle.

Step 2: Coordinate Rotation Formula

The formula to find the rotated coordinates \(X, Y\) of a point \(x, y\) rotated by an angle \phi\ is given by:\[ X = x\cos(\phi) + y\sin(\phi) \ Y = -x\sin(\phi) + y\cos(\phi) \]

Step 3: Calculate Rotated X-coordinate

Substitute the values \(x = \sqrt{2}, y = 4\sqrt{2}, \cos(45^{\circ}) = \frac{\sqrt{2}}{2}, \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\) into the formula for \X\:\[X = \sqrt{2} \cdot \frac{\sqrt{2}}{2} + 4\sqrt{2} \cdot \frac{\sqrt{2}}{2} \= 1 + 4 \= 5\]

Step 4: Calculate Rotated Y-coordinate

Now, substitute the same values into the formula for \Y\:\[Y = -\sqrt{2} \cdot \frac{\sqrt{2}}{2} + 4\sqrt{2} \cdot \frac{\sqrt{2}}{2} \= -1 + 4 \= 3\]

Step 5: Provide the Rotated Coordinates

After rotation, the new coordinates of the point are \(X, Y) = (5, 3\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a fundamental part of geometry and are used to determine the position of a point in a plane. They consist of a pair of values, usually represented as \(x\) and \(y\), which denote the horizontal and vertical distances from a defined origin. Understandably, they allow us to specify the location of a point on a two-dimensional plane easily.

• The coordinate axes are perpendicular to each other.
• The origin is where both axes meet, typically at \( (0, 0) \).

For the point \(\sqrt{2}, 4\sqrt{2}\), \( x = \sqrt{2}\)) represents the distance from the origin along the x-axis, and \(y = 4\sqrt{2}\) represents the distance along the y-axis. These two coordinates uniquely identify this point's location with respect to the Cartesian plane.
When we undertake coordinate rotation, these fixed values, \(x\) and \(y\), transform according to a given rule to produce new coordinates. This translation depends on the use of the rotation angle and some trigonometric functions.

Rotation Angle

The concept of a rotation angle is essential when it comes to rotating the coordinate plane. The rotation angle is the measure of the angle through which a figure or a point is rotated about the origin. In our exercise, the coordinate axes are rotated through 45 degrees.

• An angle of 45 degrees is a commonly used measure.
• This angle belongs to the first quadrant of a circle, providing a positive rotation.

Rotation is usually performed counter-clockwise from the positive x-axis. The angle of rotation determines the adjustment of the original coordinates to find the new coordinates.
Utilizing trigonometric functions, such as cosine and sine, for this rotation helps effectively calculate the new position of the point. Rotating can have several practical applications, such as in graphics programming, robotics, and physics simulations where changing perspectives or aligning models is crucial.

Trigonometric Functions

Trigonometric functions, such as sine (\sin) and cosine (\cos), are essential in interpreting and executing rotations within the Cartesian coordinate system. These functions are foundational in the rotation formulas that re-compute point coordinates.
To rotate a point by an angle \(\phi\), the new coordinates \(X, Y\) are calculated using:

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

In this exercise, with \(\phi = 45^{\circ}\), both cosine and sine of 45 degrees equal \(\frac{\sqrt{2}}{2}\). These values simplify the process of calculating new coordinates:

• For \(X\): Multiply and sum contributions of original \(x\) and \(y\) based on these trigonometric values.
• For \(Y\): Similarly, adjust using sine and cosine contributions.

These trigonometric functions ensure the coordinates rotate according to standard geometrical principles, maintaining the proportional relationships between points in the plane.