Problem 37 - Precalculus with Limits

Question

In Exercises 37-54, a point in rectangular coordinates is given. Convert the point to polar coordinates. \(\left(1, 1\right)\)

Step-by-Step Solution

Answer

The polar coordinates of the point (1, 1) are \((\sqrt{2}, \pi/4)\).

Step 1: Calculate r

First, calculate r using the formula \(r = \sqrt{x^2 + y^2}\). In this case, \(r = \sqrt{1^2 + 1^2} = \sqrt{2}.\)

Step 2: Calculate θ

Next, calculate \(\theta\) using the formula \(\theta = \arctan(y/x)\). In this case, \(\theta = \arctan(1/1) = \arctan(1) = \pi/4\) (or 45°), given that the point is in the first quadrant and arctan(1) is π/4 or 45°.

Step 3: Write the point in polar coordinates

Lastly, present the point in polar coordinates as \((r, \theta)\). Here, the point becomes \((\sqrt{2}, \pi/4)\).

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system for defining points in a plane using a pair of numerical values. These values are usually denoted as \(x, y\).

• The x-coordinate represents the horizontal distance of the point from the origin, which is where the axes intersect.
• The y-coordinate represents the vertical distance of the point from the origin.

Imagine plotting these coordinates on graph paper. You would move x units along the horizontal axis and y units along the vertical axis.

Rectangular coordinates are incredibly common due to their simplicity and are often the first system learned in mathematics courses. They are perfect for tasks like graphing functions or locating points on a grid.

Conversion to Polar Coordinates Explained

The conversion from rectangular coordinates to polar coordinates involves using two formulas to redefine the location of a point.

• First, calculate the distance from the origin using the formula: \(r = \sqrt{x^2 + y^2}\). This distance, r, is known as the radius.
• Second, find the angle, \(\theta\), relative to the positive x-axis, using the formula: \(\theta = \arctan(y/x)\).

The point (1, 1) in rectangular coordinates converts to \(r = \sqrt{2}\) and \(\theta = \pi/4\) radians.

This conversion process helps in tasks where distances and angles provide more intuitive insights, such as when dealing with circle equations or when integrating functions in polar coordinates.

Different Coordinate Systems

Coordinate systems are essential for describing positions on a plane or in space, allowing for various applications in navigation, physics, and engineering.

• Rectangular coordinates use perpendicular lines (axes) to find points by distance in each dimension. Usually denoted as (x, y) in 2D or (x, y, z) in 3D.
• Polar coordinates use a combination of distance from a central point (radius) and an angle to describe a point. This is often useful in scenarios involving circular symmetry.

Each system has its strengths and weaknesses, which can make one more suitable than the other depending on the situation.

Understanding these systems is crucial for effectively solving problems across different mathematical contexts, enabling smoother transformations between various forms of mathematical expressions.