Problem 5 - Precalculus Mathematics for Calculus

Question

Determine the \(X Y\)-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(3,-\sqrt{3}), \quad \phi=60^{\circ}$$

Step-by-Step Solution

Answer

The rotated point's coordinates are \((3, \sqrt{3})\).

Step 1: Understanding the Problem

We are given a point with coordinates \((3, -\sqrt{3})\) and an angle of rotation \(\phi = 60^{\circ}\). We need to calculate the new coordinates \((X,Y)\) after rotating the point by this angle around the origin.

Step 2: Using Rotation Formula

To find the new coordinates \((X, Y)\), we use the rotation transformation formulas: \[X = x \cos(\phi) - y \sin(\phi) \]\[Y = x \sin(\phi) + y \cos(\phi)\]where \((x, y)\) are the original coordinates, and \(\phi\) is the angle of rotation.

Step 3: Substituting Values into the Formula

Substitute \(x = 3\), \(y = -\sqrt{3}\), and \(\phi = 60^{\circ}\) into the formulas:\[X = 3 \cos(60^{\circ}) - (-\sqrt{3}) \sin(60^{\circ})\]\[Y = 3 \sin(60^{\circ}) + (-\sqrt{3}) \cos(60^{\circ})\]

Step 4: Calculating Trigonometric Values

Recall that \(\cos(60^{\circ}) = \frac{1}{2}\) and \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\). Substitute these values into the equations:\[X = 3 \left(\frac{1}{2}\right) - (-\sqrt{3}) \left(\frac{\sqrt{3}}{2}\right)\]\[Y = 3 \left(\frac{\sqrt{3}}{2}\right) + (-\sqrt{3}) \left(\frac{1}{2}\right)\]

Step 5: Performing the Calculations

Calculate the expressions:\[X = \frac{3}{2} + \frac{3}{2} = 3\]\[Y = \frac{3\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3}\]

Step 6: Solution Conclusion

So, the coordinates of the point after rotation are \( (3, \sqrt{3}) \).

Key Concepts

Rotation Formulas

When working with coordinate rotation, a key tool at our disposal is the rotation formulas. These formulas allow us to calculate new coordinates for a point that has been rotated around the origin by a specified angle. The formulas are derived from basic trigonometric principles and are expressed as:

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

In these equations, \((x, y)\) represents the original coordinates of the point, \((X, Y)\) are the coordinates after rotation, and \(\phi\) is the angle of rotation. This rotation is typically performed in a counterclockwise direction. Using these formulas ensures accuracy when transforming a point's position, making it a crucial concept in geometry and physics.

Trigonometric Functions

Trigonometric functions play a significant role in the calculation involved in coordinate rotation. Specifically, sine and cosine functions are used to determine the horizontal and vertical shifts of a point when rotated.Cosine of an angle \(\phi\), denoted as \(\cos(\phi)\), represents the ratio of the adjacent side to the hypotenuse of a right-angled triangle. Similarly, the sine of an angle \(\phi\), denoted as \(\sin(\phi)\), gives the ratio of the opposite side to the hypotenuse. For example, with \(\phi = 60^{\circ}\), you have:

• \(\cos(60^{\circ}) = \frac{1}{2}\)
• \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\)

These trigonometric values are fundamental when applying rotation formulas, as they determine how much the point moves horizontally and vertically due to rotation.

Coordinate Transformation

Coordinate transformation refers to the process of converting a point from one coordinate system to another. Rotation of coordinates around the origin is a classic example of such a transformation.When we rotate a point, such as \((3, -\sqrt{3})\), using an angle \(\phi = 60^{\circ}\), we alter its position based on the rotation formulas discussed earlier. This transformation implies changing the orientation of the Cartesian plane in which the point lies, thereby providing new coordinates \((3, \sqrt{3})\).The importance of this transformation lies in its wide application in dynamics, graphics, and robotics, where spatial orientation matters. It provides a mathematical way to account for the rotational movement of objects, enabling complex systems to be modeled and understood with relative ease.