Problem 5 - Algebra and Trigonometry

Question

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(0,2), \quad \phi=55^{\circ}$$

Step-by-Step Solution

Answer

The new coordinates are \((1.6384, 1.1472)\).

Step 1: Understand the problem

We are given a point with coordinates \((0, 2)\). We need to find the new coordinates of this point after the coordinate axes are rotated by \(\phi = 55^{\circ}\).

Step 2: Recall the rotation formulas

When the coordinate axes are rotated by an angle \( \phi \), the new coordinates \((x', y')\) of a point \((x, y)\) can be calculated using the formulas:\[ x' = x \cos \phi + y \sin \phi\]\[ y' = -x \sin \phi + y \cos \phi\]

Step 3: Substitute known values

Substitute \(x = 0\), \(y = 2\), and \(\phi = 55^{\circ}\) into the formulas:\[ x' = 0 \cdot \cos 55^{\circ} + 2 \cdot \sin 55^{\circ}\]\[ y' = -0 \cdot \sin 55^{\circ} + 2 \cdot \cos 55^{\circ}\]

Step 4: Calculate \(\sin 55^{\circ}\) and \(\cos 55^{\circ}\)

Using trigonometric tables or a calculator, find:\( \sin 55^{\circ} \approx 0.8192 \)\( \cos 55^{\circ} \approx 0.5736 \)

Step 5: Compute the new coordinates

Substitute \( \sin 55^{\circ} \) and \( \cos 55^{\circ} \) into the equations calculated before:\[ x' = 0 + 2 \times 0.8192 = 1.6384\]\[ y' = 0 + 2 \times 0.5736 = 1.1472\]

Step 6: Write the final answer

The new coordinates of the point after rotation are \((1.6384, 1.1472)\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, especially when dealing with angles and rotations. There are three primary trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function relates the angles of a right triangle to its sides.

• **Sine:** This function relates the opposite side of an angle to the hypotenuse, calculated as \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\).
• **Cosine:** It relates the adjacent side of an angle to the hypotenuse, expressed as \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).

Using these functions, we can find ratios for any angle. For coordinate rotations, knowing precise values for \(\sin\phi\) and \(\cos\phi\) is crucial. These values dictate how far along the x and y axis the point will move upon rotation. Calculators and trigonometric tables are commonly used to determine these values, as they involve complex calculations.

Rotation Formulas

Rotation formulas help us determine new coordinates of a point after the coordinate axes have been rotated by a certain angle, \(\phi\). If a point starts at coordinates \((x, y)\), the new coordinates \((x', y')\) can be calculated using these formulas:

• **New X-coordinate:** \(x' = x \cos \phi + y \sin \phi\). This formula tells us that the new x-coordinate is determined by both the original x and y values, filtered through the \(\cos\) and \(\sin\) of the rotation angle.
• **New Y-coordinate:** \(y' = -x \sin \phi + y \cos \phi\). Similarly, this formula calculates the new y-coordinate by a combination of the initial coordinates and the trigonometric transformations for the angle \(\phi\).

By following these formulas step by step, you can determine the precise location of any point after rotation. Whether you're working with theoretical math or real-world applications like gaming or graphics, these formulas are indispensable.

Angle of Rotation

The angle of rotation, noted as \(\phi\), is a critical factor in rotating a coordinate plane. Rotation in geometry involves moving points around a central origin directionally—usually in a circular motion—by some angle. Understanding how this works is fundamental for modifying points mathematically.
Here are several key ideas about angles of rotation:

• **Positive or Negative Rotation:** In mathematical terms, a positive angle indicates counterclockwise rotation, while a negative angle defines clockwise rotation.
• **Measure of Angles:** Angles are usually measured in degrees or radians. In this exercise, the angle was provided in degrees, which is common for geometric transformations.
• **Effect of Rotation:** The longer the angle, the greater the rotation. This means that larger angles move points further from their original positions.

Grasping these fundamentals allows you to predict how a rotation will impact positional coordinates, both visually and mathematically. Understanding angles and their role in rotations helps in seamlessly transitioning between coordinate systems. This not only has applications in learning exercises but in complex fields like animation and navigation systems as well.