Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Robot's arm Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 18 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at \((18,0)\) and then rotates through an angle of \(60^{\circ}\), what is the new location of the hand?

Step-by-Step Solution

Answer

The new location of the robot's hand is at (9, 9√3).

Step 1: Understanding the Problem

The robot's arm is initially at a length of 18 inches along the positive x-axis, represented by the point (18, 0). The arm rotates through an angle of \(60^{\circ}\). We need to find the new coordinates of the hand after rotation.

Step 2: Representing the Angle in Radians

To use trigonometric functions effectively, convert the angle from degrees to radians. Use the formula: \[\text{Radians} = \frac{\pi}{180} \times \text{degrees} = \frac{\pi}{180} \times 60 = \frac{\pi}{3}\]

Step 3: Calculating New Coordinates Using Rotation Formulas

Using the rotation formulas for point (x, y) rotated by θ radians around the origin:\[x' = x \cdot \cos(\theta) - y \cdot \sin(\theta) \y' = x \cdot \sin(\theta) + y \cdot \cos(\theta)\]Substitute \((x, y) = (18, 0)\) and \(\theta = \frac{\pi}{3}\):\[x' = 18 \cdot \cos\left(\frac{\pi}{3}\right) - 0 \cdot \sin\left(\frac{\pi}{3}\right) = 18 \cdot \frac{1}{2} = 9\y' = 18 \cdot \sin\left(\frac{\pi}{3}\right) + 0 \cdot \cos\left(\frac{\pi}{3}\right) = 18 \cdot \frac{\sqrt{3}}{2} = 9\sqrt{3}\]

Step 4: Writing the Final Answer

After rotation, the new position of the robot's hand is at point \((9, 9\sqrt{3})\).

Key Concepts

Coordinate Geometry

In coordinate geometry, we map mathematical concepts to a grid system using points defined by coordinates. This grid, also known as the Cartesian plane, consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is specified as an ordered pair \((x, y)\), where 'x' represents the horizontal displacement from the origin and 'y' denotes the vertical displacement.

When analyzing movements like the rotation of a robot's arm, the origin (0,0) serves as the pivot point. This makes it simpler to determine the new coordinates of rotated points using mathematical transformations. For example, the robot's initial hand position at (18, 0) signifies a point 18 units away on the x-axis from the origin.

Understanding the grid is crucial for various applications beyond robotics, such as physics and engineering, where spatial relationships are modeled and analyzed using these principles.

Robotics

Robotics involves the design, construction, operation, and use of robots, typically using a combination of electrical engineering, computer science, and mechanical engineering. One fundamental aspect of a robot's physical design is its ability to manipulate objects and navigate through its environment efficiently.

The robot's arm example illustrates how robotics uses coordinate geometry to determine the position of various parts during operation. The arm's ability to rotate is central to many robotic tasks, such as picking and placing items, assembling parts, or moving through constrained environments.

Robotics relies heavily on understanding both geometry and mechanics. By applying transformations like rotations, it enables robots to perform complex, precise tasks that mimic human actions or handle tasks that are dangerous for humans to undertake.

As robotics technology advances, programming and designing these robotic movements using geometry becomes increasingly sophisticated, opening up new possibilities across different fields.

Angle Rotation

Angle rotation is a fundamental concept in mathematics and physics, vital for understanding any system with moving parts. When an object rotates around a point, it changes its orientation and position. This movement can be analyzed using trigonometric functions and rotational transformations.

In our robotics example, the robotic arm rotates about the origin by an angle of \(60^\circ\). To determine its new location after rotation, we need to understand how rotation affects its coordinates. The general formulas for rotating a point \((x, y)\) by an angle \(\theta\) about the origin are:\[ x' = x \cdot \cos(\theta) - y \cdot \sin(\theta) \] \[ y' = x \cdot \sin(\theta) + y \cdot \cos(\theta) \]

These equations are indispensable in robotics for calculating new positions of parts without physical trial and error. They also provide the mathematical foundation to simulate and predict movements in software, contributing to the development of algorithms for automation and navigation.

Radian Conversion

In the world of trigonometry and geometry, angles can be measured in degrees or radians. Radians provide a more natural mathematical language when dealing with periodic functions and rotations. Conversion between these two measurements is essential for integrating trigonometric functions into various calculations.

The conversion from degrees to radians is straightforward: multiply the number of degrees by \(\frac{\pi}{180}\). For our robotic arm, a \(60^\circ\) angle in degrees converts to \(\frac{\pi}{3}\) radians, as shown by the formula:

\[ \text{Radians} = \frac{\pi}{180} \times \text{degrees} = \frac{\pi}{180} \times 60 = \frac{\pi}{3} \]

Using radians allows for a smoother integration into equations involving trigonometric functions, which are pivotal in solving problems related to rotations and oscillations. Understanding this conversion is fundamental in fields requiring precise angle measurements, such as physics, astronomy, and, as seen in this exercise, robotics.