Problem 4 - Precalculus: Mathematical for Calculus

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 the Problem

We need to find new coordinates of the point \((2,0)\) after the coordinate axes are rotated by \(15^{\circ}\). The formula for the new coordinates \((X, Y)\) after rotation by \(\phi\) is given as: \[ X = x \cos(\phi) + y \sin(\phi) \]\[ Y = -x \sin(\phi) + y \cos(\phi) \] where \((x, y)\) are the original coordinates.

Step 2: Substitute the Given Values

In the formula, substitute \(x = 2\), \(y = 0\), and \(\phi = 15^{\circ}\). The trigonometric values we need are: \[ \cos(15^{\circ}) \approx 0.9659 \]\[ \sin(15^{\circ}) \approx 0.2588 \].

Step 3: Calculate the New X-Coordinate

Plug in the values into the formula for \(X\):\[ X = 2 \cdot 0.9659 + 0 \cdot 0.2588 = 1.9318 \].

Step 4: Calculate the New Y-Coordinate

Substitute the values into the formula for \(Y\):\[ Y = -2 \cdot 0.2588 + 0 \cdot 0.9659 = -0.5176 \].

Step 5: Present the Final Result

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

Key Concepts

Trigonometric Values

Trigonometric values play a pivotal role in solving coordinate rotation problems. They help us determine the new coordinates of a point when the coordinate axes are rotated by an angle \(\phi\).

• The two fundamental trigonometric functions used for rotations are the sine \(\sin(\phi)\) and cosine \(\cos(\phi)\).
• These functions give us the relationship between the angle of rotation and the coordinates of the point.

For instance, when rotating a point \((2,0)\) by \(15^{\circ}\), we need the trigonometric values:

• \(\cos(15^{\circ})\) is approximately 0.9659
• \(\sin(15^{\circ})\) is approximately 0.2588

These values help calculate the new position of the point in the coordinate system. When you know the trigonometric values, you can easily plug them into the rotation formula to find the new coordinates.

Rotation Formula

The rotation formula is essential for identifying how coordinates change when the axes are rotated by a certain angle. This formula gives us an easy method to compute the new coordinates \((X, Y)\) from the original ones \((x, y)\).The formula is expressed as:- \(X = x \cos(\phi) + y \sin(\phi)\)- \(Y = -x \sin(\phi) + y \cos(\phi)\)When applying this formula:

• Substitute the specific trigonometric values for the angle \(\phi\).
• Use the original coordinates in place of \(x\) and \(y\).

In our example, with \(x = 2\) and \(y = 0\) and \(\phi = 15^{\circ}\), we use:

• \(X = 2 \cdot 0.9659 + 0 \cdot 0.2588 = 1.9318\)
• \(Y = -2 \cdot 0.2588 + 0 \cdot 0.9659 = -0.5176\)

Through this formula, you can understand how any point will shift after rotating the axes by a specified angle.

Angle Conversion

Angle conversion is a crucial step when dealing with rotations and trigonometric functions. Before utilizing trigonometric values, confirm that the angle is in a suitable unit, which is usually degrees or radians.

• By default, many calculations and scientific calculators use radians.
• For our example, \(15^{\circ}\) must ensure we use degrees to fetch the right trigonometric values.

To convert degrees to radians, use:\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]In some scenarios, especially when using software tools, conversions are vital. For instance, converting \(15^{\circ}\) to radians would involve:

• \(15 \times \frac{\pi}{180} = \frac{\pi}{12}\)

Despite the conversion, our task remains simplified as trigonometric tables or calculators usually provide direct values for common angles regardless of their unit. Always check the requirements of your task before proceeding with calculations.