Problem 6 - Precalculus Mathematics 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 (1.93, -0.52).

Step 1: Understand the Problem

We need to find the new coordinates of the point \(2, 0\) after rotating the coordinate axes by \(\phi = 15^{\circ}\).

Step 2: Know the Formula

To solve this, use the rotation formula for transformation of coordinates: \\(x' = x \cos(\phi) + y \sin(\phi)\), \\(y' = -x \sin(\phi) + y \cos(\phi)\).\.

Step 3: Plug in Known Values

Insert the point \(x = 2, y = 0\) and \(\phi = 15^{\circ}\) into the formulas: \\(x' = 2 \cos(15^{\circ}) + 0 \sin(15^{\circ})\), \\(y' = -2 \sin(15^{\circ}) + 0 \cos(15^{\circ})\).

Step 4: Calculate New Coordinates

Determine \(\cos(15^{\circ}) = \sqrt{6}/4 + \sqrt{2}/4 \approx 0.9659 \\and \\sin(15^{\circ}) = \sqrt{6}/4 - \sqrt{2}/4 \approx 0.2588\). Plug these into each transformation formula: \\(x' = 2 \times 0.9659 = 1.9318\), \\(y' = -2 \times 0.2588 = -0.5176\).

Step 5: Write the Final Answer

The new coordinates of the point are approximately \( (1.93, -0.52) \) when rounded to two decimal places.

Key Concepts

Trigonometric Functions

Trigonometric functions are crucial mathematical tools used to find relationships between angles and sides of triangles. When dealing with coordinate transformation, as in the given problem, we particularly use the sine and cosine functions. These functions are defined as follows:

• The sine function, \( \sin(\theta) \), is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• The cosine function, \( \cos(\theta) \), is the ratio of the adjacent side to the hypotenuse.

In our exercise, we utilized these functions to rotate the coordinates of a point around the origin. Given an angle \( \phi \), we can accurately calculate the requirement for sine and cosine, deriving the coordinate changes necessary in the rotation transformation. Remember, the values of these functions for certain key angles are often memorized or can be derived using trigonometric identities.

Transformation of Coordinates

Transformation of coordinates refers to the process of changing the viewpoint or reference axes in which a particular point or object is described. In the exercise, we perform a type of transformation known as "rotation of axes."

The general formulas for this kind of transformation are given as:

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

These equations effectively compute the new coordinates \((x', y')\) after rotating the axes by an angle \(\phi\). This mathematical process allows us to see how the given point \((2,0)\) shifts when the coordinate grid itself is "twisted" around the origin by \(15^{\circ}\). Utilizing this transformation, not only do we acquire different new points, but it can also provide insights into understanding geometrical properties invariant under rotation.

Angle Rotation

Angle rotation, in geometry, refers to the movement of a shape around a central point. In coordinate geometry, this involves pivoting the entire coordinate grid by a specified angle. For this exercise, we rotated the axes by \(15^{\circ}\).

It's important to grasp that rotation does not change the shape or size of the object, only its orientation in space. This is why the point's distance from the origin remains constant after rotation.

• For positive rotation angles, rotation is in the counterclockwise direction.
• For negative rotation angles, it’s in the clockwise direction.

Understanding angle rotation is vital in contexts where directions are pivotal, such as physics problems involving motion, computer graphics, and even robotics.

Precalculus Mathematics

Precalculus mathematics prepares students for calculus studies by introducing them to concepts such as functions, sets, and sequences. A key part of precalculus involves understanding geometric transformations using algebraic methods, like coordinate rotation. This ensures a deep foundation in both geometry and algebra.

Coordinate rotation in precalculus involves the application of trigonometry to practical problems, as seen in this exercise. Students learn to:

• Apply trigonometric identities and functions.
• Perform transformations in the coordinate plane efficiently.
• Work with angles and their implications on geometric figures.

By practicing these concepts, students develop a strong mathematical skillset enabling a smooth transition to calculus and advanced topics like vector calculus and linear algebra.