Problem 4 - Precalculus: Mathematics for Calculus

Question

Determine the XY-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 after rotation are approximately \\((-\sqrt{3} - 0.5, -1 + \frac{1.732}{2})\\)."}

Step 1: Understand the Rotation Transformation

When a point \(x,y\) is rotated about the origin by an angle \(\phi\), its new position in the coordinate plane can be described as \( (x', y') \) using the rotation formulas:\[ x' = x \cdot \cos\phi - y \cdot \sin\phi \]\[ y' = x \cdot \sin\phi + y \cdot \cos\phi \]

Step 2: Convert Angle to Radians

The rotation angle is given in degrees. Convert \(30^\circ\) to radians for calculation purposes, as trigonometric functions in most calculations use radians. The conversion is:\[ 30^\circ = \frac{30 \cdot \pi}{180} = \frac{\pi}{6} \]

Step 3: Calculate New X-coordinate

Use the given point \((-2,1)\) and the rotation angle \(\phi = \frac{\pi}{6}\) in the first rotation formula to find \(x'\):\[ x' = -2 \cdot \cos\left(\frac{\pi}{6}\right) - 1 \cdot \sin\left(\frac{\pi}{6}\right) \]Substitute the values of cos and sin:\[ x' = -2 \cdot \left(\frac{\sqrt{3}}{2}\right) - 1 \cdot \left(\frac{1}{2}\right) = -\sqrt{3} - \frac{1}{2} \]

Step 4: Calculate New Y-coordinate

Use the given point and rotation angle in the second rotation formula to find \(y'\):\[ y' = -2 \cdot \sin\left(\frac{\pi}{6}\right) + 1 \cdot \cos\left(\frac{\pi}{6}\right) \]Substitute the values of cos and sin:\[ y' = -2 \cdot \left(\frac{1}{2}\right) + 1 \cdot \left(\frac{\sqrt{3}}{2}\right) = -1 + \frac{\sqrt{3}}{2} \]

Step 5: Simplify Result

Simplify the expressions for \(x'\) and \(y'\) to find the new coordinates:- For \(x'\), \(-\sqrt{3} - \frac{1}{2}\)- For \(y'\), \(-1 + \frac{\sqrt{3}}{2}\)

Key Concepts

Coordinate Geometry

Coordinate Geometry is like a map for understanding where points are located on a plane. Think of it as a way to give every point an address using numbers. In a usual coordinate system, we use two numbers for any point: an X-coordinate (the horizontal position) and a Y-coordinate (the vertical position).

These coordinates are based on a rectangular grid where there are two main lines: the X-axis (going left and right) and the Y-axis (going up and down). Together, these axes form the origin, which is at the '0' point on both axes.

• Positive X-coordinates are to the right of the origin.
• Negative X-coordinates are to the left of the origin.
• Positive Y-coordinates are above the origin.
• Negative Y-coordinates are below the origin.

When we talk about transforming coordinates, like in rotation, we're changing the point's address using specific rules. This is why Coordinate Geometry is so useful in understanding movements and transformations on a plane.

Trigonometric Functions

Trigonometric Functions help us understand angles and sides in triangles, and they play a big role in the rotation of points in the coordinate plane. In the context of the rotation transformation, we primarily use sine \((\sin)\) and cosine \((\cos)\). These functions tell us about the relationships between different parts of a right triangle.

• The cosine of an angle gives us the ratio of the adjacent side to the hypotenuse in a right triangle.
• The sine of an angle gives us the ratio of the opposite side to the hypotenuse.

When rotating a point, trigonometric functions help calculate the new coordinates based on the rotation angle. For example, in rotation, \(x'\) and \(y'\) are determined by the \((\cos\phi)\) and \(\sin\phi\) values, respectively. This is essential to retain the structure of the point in its new position.

Radians to Degrees Conversion

Radians and degrees are two ways of measuring angles. While degrees are more familiar (like in a circle with 360°), radians are often used in mathematical calculations, especially involving trigonometric functions.

To convert from degrees to radians, it's important to remember that the complete circle is \(2\pi\) radians, which is equivalent to 360°. Therefore, to convert any angle from degrees to radians, the formula is:

• \(\text{{Radians}} = \text{{Degrees}} \times \frac{\pi}{180}\).

So for \(30^\circ\), using this formula, we get the conversion as \(\frac{30 \times \pi}{180} = \frac{\pi}{6}\).

Using radians can make calculations with trigonometric functions more straightforward, especially when transformations like rotations are involved.

Transformation Formulas

Transformation Formulas provide the rules for changing points on a coordinate plane. In our case, a rotation transformation is what we're focusing on. It's like spinning around the origin on a fixed point, given by a specific angle, here denoted as \(\phi\).
The core formulas for a rotation transformation are:

• New X: \(x' = x \cdot \cos\phi - y \cdot \sin\phi\)
• New Y: \(y' = x \cdot \sin\phi + y \cdot \cos\phi\)

Here:

• \(x\) and \(y\) are the original coordinates.
• \(x'\) and \(y'\) are the new coordinates after rotation.
• \(\cos\phi\) and \(\sin\phi\) are the cosine and sine of the rotation angle.

These formulas ensure that the new position of a point after rotation retains its relative distances but changes its direction as defined by the angle \(\phi\). Understanding these formulas is crucial for solving problems in rotation transformation effectively.