Problem 4 - Algebra and Trigonometry

Question

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(2,0), \quad \phi=15^{\circ}$$

Step-by-Step Solution

Answer

The new coordinates are approximately \((1.9318, -0.5176)\).

Step 1: Understand Rotation of Axes

When the coordinate axes are rotated by an angle \(\phi\), the new \(X\) and \(Y\) coordinates \((X,Y)\) of the point \((x,y)\) can be found using the transformation equations: \[X = x \cos(\phi) + y \sin(\phi)\] \[Y = -x \sin(\phi) + y \cos(\phi)\]

Step 2: Convert Degrees to Radians

We need to convert the given angle \(\phi = 15^{\circ}\) into radians since trigonometric functions in calculus are generally calculated in radians. \[\phi = 15^{\circ} \times \frac{\pi}{180} = \frac{\pi}{12}\]

Step 3: Calculate New X-Coordinate

Substitute \(x = 2\), \(y = 0\), and \(\phi = \frac{\pi}{12}\) into the equation for \(X\): \[X = 2 \cos\left(\frac{\pi}{12}\right) + 0 \sin\left(\frac{\pi}{12}\right) = 2 \cos\left(\frac{\pi}{12}\right)\] Calculate \(\cos\left(\frac{\pi}{12}\right)\) and find \(X\). \(\cos\left(\frac{\pi}{12}\right) \approx 0.9659\). Thus, \[X \approx 2 \times 0.9659 = 1.9318\]

Step 4: Calculate New Y-Coordinate

Substitute \(x = 2\), \(y = 0\), and \(\phi = \frac{\pi}{12}\) into the equation for \(Y\): \[Y = -2 \sin\left(\frac{\pi}{12}\right) + 0 \cos\left(\frac{\pi}{12}\right) = -2 \sin\left(\frac{\pi}{12}\right)\] Calculate \(\sin\left(\frac{\pi}{12}\right)\) and find \(Y\). \(\sin\left(\frac{\pi}{12}\right) \approx 0.2588\). Thus, \[Y \approx -2 \times 0.2588 = -0.5176\]

Step 5: Final XY-Coordinates

The \(XY\)-coordinates of the point after a \(15^{\circ}\) rotation of the axes are approximately \((1.9318, -0.5176)\).

Key Concepts

Coordinate Transformation

When we talk about coordinate transformation, we're discussing how to change the way we look at points on a plane by altering the coordinate system. Imagine turning a piece of paper to get a new view of the drawing on it; this is akin to rotating axes.

In the context of this exercise, we change from (x, y) coordinates to new (X, Y) coordinates by rotating the axes by a specified angle, \( \phi \). This rotation affects the original coordinates and enables us to view points relative to a different orientation.

To perform this transformation, we use the following equations:

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

These equations allow us to calculate the new coordinates in the rotated system. It's a vital tool in fields like physics, engineering, and computer graphics.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a fundamental role in coordinate transformation. They help us relate angles to lengths, which is key when changing from one set of axes to another.

The cosine function \( \cos(\phi) \) helps us determine the new X-coordinate by scaling the x-component of the point. Similarly, the sine function \( \sin(\phi) \) is crucial for both adjusting the X and Y coordinates by incorporating the angle \( \phi \).

In our example:

• \( X = x \cos(\phi) + y \sin(\phi) \) involves both trigonometric functions to calculate X.
• \( Y = -x \sin(\phi) + y \cos(\phi) \) utilizes them to find Y.

Understanding these functions and how they work is essential for performing transformations accurately, ensuring precise real-world applications.

Radians Conversion

Converting degrees to radians is a crucial step, particularly because most trigonometric function calculations are done in radians. Degrees are familiar, but radians are a mathematical standard.

To convert an angle \( \phi \) given in degrees to radians, we use the conversion factor \( \frac{\pi}{180} \). This means:

• \( \phi = 15^{\circ} \)
• Convert with \( \phi = 15^{\circ} \times \frac{\pi}{180} \)
• Result is \( \phi = \frac{\pi}{12} \)

Understanding this conversion ensures that calculations align with how trigonometric functions operate by default, allowing for seamless application across different mathematical problems.

Algebra

Algebra helps us handle the mathematical expressions and transformations during coordinate rotation by letting us manage equations systematically.

In our scenario, once we have values for \( \cos(\phi) \) and \( \sin(\phi) \), we substitute these into the equations to find the new coordinates \( X \) and \( Y \).

The steps are straightforward:

• Find \( \cos\left(\frac{\pi}{12}\right) \approx 0.9659 \), and \( X = 2 \times 0.9659 = 1.9318 \)
• Calculate \( \sin\left(\frac{\pi}{12}\right) \approx 0.2588 \), and \( Y = -2 \times 0.2588 = -0.5176 \)

These algebraic manipulations render a clear result: new coordinates \((1.9318, -0.5176)\). Such processes are the building blocks of solving similar mathematical problems and are often encountered in higher-level studies in various fields.