Problem 7 - Precalculus with Limits

Question

In Exercises 5-12, the \(x'y'\)-coordinate system has been rotated \(\theta\) degrees from the \(xy\)-coordinate system. The coordinates of a point in the \(xy\)-coordinate system are given. Find the coordinates of the point in the rotated coordinate system. \(\theta=30^{\circ}, (1, 3)\)

Step-by-Step Solution

Answer

The coordinates of the point in the rotated system are given by (\(x'\), \(y'\)).

Step 1 - Conversion of Degree into Radians

Angular measurement in most mathematical functions is expressed in radians. So, the given rotation angle from degrees to radians must be converted. The conversion if

Step 2 - Identify the Original Coordinates

The original \(xy\)-coordinates are given as (1, 3). So, \(x = 1\) and \(y = 3\).

Step 3 - Use the Rotation Formula

To find the new coordinates on the rotated system, the rotation formulas \(x'= x \cos(\theta) - y \sin(\theta)\) and \(y'= x \sin(\theta) + y \cos(\theta)\) are used. Plugging \(x = 1\), \(y = 3\), and \(\theta\) into these formulas, calculate \(x'\) and \(y'\).

Step 4 - Calculate the New Coordinates

Compute the numerical value for \(x'\) and \(y'\) to get the coordinates of the point in the rotated system.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It plays a vital role in understanding various mathematical problems, including the coordinate rotation we deal with here. When rotating a point around the origin by an angle \(\theta\), trigonometry helps us use sine and cosine functions to find the new positions of points in a rotated coordinate system.

• Cosine (cos): Represents the adjacent side over hypotenuse in a right-angled triangle.
• Sine (sin): Represents the opposite side over hypotenuse in a right-angled triangle.

By utilizing these functions in the rotation formula, we can calculate the new positions of points in a rotated coordinate plane without physically moving the points. This concept is crucial since it enables transformations and graphs in multiple fields like physics and engineering.

Coordinate Systems

Coordinate systems are a framework that defines how locations are determined in a plane or space. The most common is the Cartesian coordinate system, represented by two perpendicular axes: the \(x\)-axis (horizontal) and the \(y\)-axis (vertical).

• Original Coordinates (x,y): These are the given initial coordinates before rotation.
• Rotated Coordinates (x',y'): These are the new positions of coordinates after applying a rotation.

When you perform a coordinate rotation, you essentially shift the point without altering its distance from the origin. The classic formulas for rotation ensure the preservation of these distances and alter only the bearing. Thus, understanding coordinates and how they shift with rotations forms the foundation for navigation and control systems.

Radians Conversion

Radians are an alternative way of measuring angles instead of degrees, and are frequently used in mathematical calculations. One full rotation around a circle equals \(2\pi\) radians, which corresponds to \(360^\circ\). Thus, converting degrees to radians is crucial since most mathematical functions, including trigonometric functions, inherently use radians.
To convert an angle from degrees to radians, use the formula:\[ ext{Radians} = ext{Degrees} \times \left(\frac{\pi}{180}\right)\] In our exercise, a \(30^\circ\) angle becomes:\[30 \times \left(\frac{\pi}{180}\right) = \frac{\pi}{6} \, \text{radians}\]By ensuring we work with radians, we allow ourselves to use these angles directly in trigonometric functions for calculating new coordinates in a rotated system. This conversion forms a fundamental step in many engineering and physics-based calculations.