Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(0,-5)$$

Step-by-Step Solution

Answer

The polar coordinates of the point (0, -5) are (5, \(3\pi / 2\)) and (5, \(7\pi/2\)).

Step 1: Plotting the Point

The point given in rectangular coordinates is (0, -5). This means that it is 0 units to the right of the origin and 5 units below the origin. Therefore, it lies on the negative y-axis.

Step 2: Converting Rectangular Coordinates to Polar Coordinates

A point in polar coordinates is represented as (r, \(\theta\)), where r represents the distance from the origin and \(\theta\) represents the angle from the positive x-axis. To convert the given rectangular coordinates to polar coordinates, we can use the formulas: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(y/x)\). However, as our point lies on the y-axis, \(\theta\) will be either \(\pi / 2\) or \(3\pi / 2\) depending on the sign of y. Since y is negative, our \(\theta\) in this case will be \(3\pi / 2\). Also, r is the distance from the origin, so it is always positive. Thus, \(r = 5\). Thus, the polar equivalent of (0, -5) is (5, \(3\pi / 2\)).

Step 3: Finding Another Set of Polar Coordinates

We can find another polar coordinate equivalent by adding \(2\pi\) to \(\theta\) to get the coordinates (5, \(7\pi/2\)) which also falls within the specified range.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way of positioning points in a plane using two values. These values correspond to the point's horizontal and vertical distances from a reference point, known as the origin. Imagine a graph where the horizontal line is labeled x, and the vertical line is labeled y. These lines intersect at a point called the origin, marked as (0,0).

• The x-value indicates the position along the horizontal axis. Positive values move to the right, and negative values move to the left.
• The y-value indicates the position along the vertical axis. Positive values move upwards, and negative values move downwards.

For example, the point (0, -5) means that you have moved 0 units on the x-axis and 5 units down the y-axis. This placement means that the point is directly on the negative y-axis.

Angle Measurement

In the context of polar coordinates, angle measurement becomes crucial. The angle, denoted as \(\theta\), defines the direction of your point relative to the positive x-axis. Angles in polar coordinates are usually measured in radians.

• One complete revolution around the origin corresponds to an angle of \(2\pi\) radians.
• Angles measured in the counter-clockwise direction from the positive x-axis are positive.
• For points directly on the y-axis, the measurement is \(\pi/2\) for the positive y-axis and \(3\pi/2\) for the negative y-axis.

In the original problem, our point (0, -5) lies directly on the negative y-axis, hence the angle \(\theta\) is \(3\pi/2\).

Coordinate Conversion

Converting between rectangular and polar coordinates involves some mathematical operations that help represent the same point in either system. To achieve this, two main formulas are utilized:

• The magnitude \(r\), which is calculated as \( r = \sqrt{x^2 + y^2} \). This formula finds the distance from the origin to the point.
• The angle \(\theta\), determined by \( \theta = \tan^{-1}(y/x) \), helps identify the point's direction relative to the positive x-axis. However, when the point lies on an axis, a simpler approach is used based on known angles.

In the case of the point (0, -5) in rectangular coordinates, converting to polar coordinates is straightforward. As it lies directly on an axis, we identify \(\theta\) as \(3\pi/2\), and the distance is \(r = 5\), resulting in the polar representation (5, \(3\pi/2\)).

Trigonometry Concepts

Trigonometry plays a significant role in converting and understanding polar coordinates. It provides the necessary relationships between angles and distances.

• Trigonometric identities such as sine, cosine, and tangent are often used. In polar coordinates, sine and cosine can determine a point’s position.
• For points on axes, angles are identified using known trigonometric angles like 0, \(\pi/2\), \pi\, and \(3\pi/2\).
• Tangent, however, mostly comes into play when identifying \(\theta\) in conversion operations with non-axis points.

With (0, -5), trigonometry helps clarify that since the y-value is negative and x is zero, the \(\theta\) for points on the negative y-axis directly corresponds to \(3\pi/2\). Using trigonometric methods simplifies this process and ensures accurate coordinate conversion.