Problem 40 - Precalculus with Limits

Question

In Exercises 37-54, a point in rectangular coordinates is given. Convert the point to polar coordinates. \(\left(-4, -4\right)\)

Step-by-Step Solution

Answer

The polar coordinates of the point \((-4, -4)\) are \((4\sqrt{2}, 225°)\) or \((4\sqrt{2}, 5\pi/4)\) in radians.

Step 1: Calculate distance from origin (polar radius)

The polar radius, often denoted as \( r \), can be calculated from rectangular coordinates using the Pythagorean theorem, i.e. \( r = \sqrt{x^2 + y^2} \). Substituting in the given x and y values, we get \( r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2} \).

Step 2: Calculate polar angle

The angle, often denoted as \( \theta \), is the angle formed by the positive x-axis and the radius. It can be found using the tangent function: \( \theta = tan^{-1}(y/x) \). We begin by substituting the y and x values: \( \theta = tan^{-1}(-4/-4) = tan^{-1}(1) \). The angle corresponding to tan(1) is 45° in the first quadrant. However, our point lies in the third quadrant. Hence, our angle should be 180°+45° = 225°. Note that if we are using radians, the corresponding radian measure of 225° is \( 5\pi/4 \).

Step 3: Present the final answer

The converted polar coordinates are expressed as \((r, \theta)\), where r is the radius and \(\theta\) is the angle. We substitute the values we've obtained to get the final answer.

Key Concepts

Rectangular Coordinates

Rectangular Coordinates are a common way to describe the position of a point in space using two numerical values: \((x, y)\). You can think of these as the point's address on a grid, with \(x\) indicating how far right or left you move from the origin (the center point, \(0, 0\)), and \(y\) indicating how far up or down you move.

These coordinates are very intuitive when working in a Cartesian plane, such as graphing lines or shapes. For example, the point \((-4, -4)\) tells us to move 4 units left and 4 units down from the origin. This is a very precise way to locate a point in two-dimensional space.

• The \(x\)-coordinate tells the horizontal direction (positive means right, negative means left).
• The \(y\)-coordinate tells the vertical direction (positive means up, negative means down).

Understanding rectangular coordinates is crucial because they form the basis from which we convert to polar coordinates.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle used to calculate distances in geometry. It provides a simple formula: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse of a right triangle, and \(a\) and \(b\) are the other two sides.

In the context of converting coordinates, \(a\) and \(b\) relate directly to the \(x\) and \(y\) axes of a point in rectangular coordinates. The theorem helps us find the polar radius \(r\), which is the distance of the point from the origin.

• Use \(x\) and \(y\) as the triangle's legs to find \(r\).
• Calculate \(r\) using \(r = \sqrt{x^2 + y^2}\).

In our example, converting \((-4, -4)\), results in \(r = 4\sqrt{2}\). This uses the Pythagorean method to ensure accuracy.

Polar Radius

The Polar Radius, usually denoted by \(r\), represents the distance from the origin to the point in polar coordinates. It is akin to measuring the length of an invisible line drawn directly from the origin to the point.

You calculate the polar radius using the Pythagorean Theorem formula: \(r = \sqrt{x^2 + y^2}\). This converts the initial rectangular coordinates into a fundamental component of polar coordinates.

• The formula \(r = \sqrt{(-4)^2 + (-4)^2}\) evaluates the distance.
• Calculation gives \(r = \sqrt{32} = 4\sqrt{2}\).

This radius is crucial in representing the size of the radius in polar coordinates, providing the point's distance from the origin.

Tangent Function

The Tangent Function is an important function in trigonometry used to determine angles. For coordinate conversion, it helps find the polar angle \(\theta\).

By using the inverse tangent formula \(\theta = \tan^{-1}(\frac{y}{x})\), you can determine the angle between the positive x-axis and the line connecting the origin to the point. It gives precise angular measurements for converting rectangular coordinates to polar coordinates.

• Substitute values to get \(\theta = \tan^{-1}(\frac{-4}{-4})\).
• This results in \(\theta = \tan^{-1}(1)\), which initially identifies a 45° angle.

Since the original point is in the third quadrant, adjustments are made to find the accurate angle measure.

Polar Angle

The Polar Angle, denoted as \(\theta\), is the measure of the angle relative to the positive x-axis, displayed counter-clockwise. This angle is crucial for expressing a point's position in polar coordinates.

Calculating the polar angle requires understanding its position in the coordinate plane's quadrants. In our example, the point \((-4, -4)\) lies in the third quadrant. Hence, adjustments are needed after the initial tangent result.

• Begin with \(\theta = \tan^{-1}(1)\), resulting in 45°.
• Add 180° for third quadrant placement, so \(\theta = 225°\).
• Alternatively, express the angle in radians, as \(\theta = \frac{5\pi}{4}\).

Understanding the polar angle ensures clear communication of a point’s direction in polar coordinates.