Question

Plot the points with polar coordinates \(\left(2, \frac{\pi}{6}\right)\) and \(\left(-3,-\frac{\pi}{2}\right) .\) Give two alternative sets of coordinate pairs for both points.

Short answer

Answer: The original points (2, π/6) and (-3, -π/2) can alternatively be represented as (-2, 7π/6), (-2, -5π/6), (3, π/2), and (-3, 3π/2) respectively.

Step-by-step solution

Convert the polar coordinates to Cartesian coordinates

To plot the points, it will be helpful to first convert the polar coordinates to Cartesian coordinates. Recall that we can use the following equations to make this conversion: \(x = r \cos{\theta}\) \(y = r \sin{\theta}\) Now, we can compute the Cartesian coordinates for each point: For the point \((2, \frac{\pi}{6})\), we have: \(x = 2 \cos{\frac{\pi}{6}} = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}\) \(y = 2 \sin{\frac{\pi}{6}} = 2 \times \frac{1}{2} = 1\) So, its Cartesian coordinates are \((\sqrt{3}, 1)\). For the point \((-3, -\frac{\pi}{2})\), we have: \(x = -3 \cos{(-\frac{\pi}{2})} = -3 \times 0 = 0\) \(y = -3 \sin{(-\frac{\pi}{2})} = -3 \times (-1) = 3\) So, its Cartesian coordinates are \((0, 3)\).

Plot the points

Now that we have the Cartesian coordinates for each point, we can plot them on a Cartesian plane. The point \((\sqrt{3}, 1)\) is in the first quadrant, and the point \((0, 3)\) is on the positive y-axis.

Find alternative coordinate pairs

To find alternative sets of coordinate pairs, we can change the value of r and theta for each point while keeping their Cartesian coordinates the same. For the point \((2, \frac{\pi}{6})\): Alternative Set 1: \(r_1 = -2\) \(\theta_1 = \frac{7\pi}{6}\) Alternative Set 2: \(r_2 = -2\) \(\theta_2 = -\frac{5\pi}{6}\) For the point \((-3, -\frac{\pi}{2})\): Alternative Set 1: \(r_1 = 3\) \(\theta_1 = \frac{\pi}{2}\) Alternative Set 2: \(r_2 = -3\) \(\theta_2 = \frac{3\pi}{2}\) In conclusion, the original points \((2, \frac{\pi}{6})\) and \((-3, -\frac{\pi}{2})\) can alternatively be represented as \((-2, \frac{7\pi}{6})\), \((-2, -\frac{5\pi}{6})\), \((3, \frac{\pi}{2})\), and \((-3, \frac{3\pi}{2})\) respectively.

Key concepts

Cartesian Coordinates

In mathematics, Cartesian coordinates provide a way to identify the location of a point on a plane using two numbers. This system is often used because it is straightforward and easy to understand.

Each point on the Cartesian plane is described by a pair of numbers:

• The value along the x-axis (horizontal direction)
• The value along the y-axis (vertical direction)

For example, the point \( (\sqrt{3}, 1) \) has an x-coordinate of \( \sqrt{3} \) and a y-coordinate of 1.

This system, pioneered by René Descartes, offers a way to precisely locate points without ambiguity. It is an essential tool in mathematics, physics, engineering, and various aspects of everyday life, like GPS technology.

Quadrants

The Cartesian coordinate system is divided into four regions known as quadrants. These quadrants help in categorizing the location of points based on the sign of their coordinates.

• First Quadrant: Both x and y coordinates are positive (e.g., \( (\sqrt{3}, 1) \)).
• Second Quadrant: x is negative, and y is positive.
• Third Quadrant: Both x and y coordinates are negative.
• Fourth Quadrant: x is positive, and y is negative.

Additionally, points that lie directly on the x-axis or y-axis are not part of any quadrant, like the point \( (0, 3) \) which is on the positive y-axis.

Understanding quadrants is key in graphing techniques and interpreting data trends.

Conversion Equations

Conversion equations are employed to transition between different coordinate systems, such as from polar coordinates to Cartesian coordinates.

In polar coordinates, a point is defined by:

• The distance from the origin (r)
• The angle from the positive x-axis (\( \theta \))

To convert polar coordinates \( (r, \theta) \) to Cartesian coordinates \( (x, y) \):

• Use \( x = r \cos{\theta} \)
• Use \( y = r \sin{\theta} \)

For instance, converting \( (2, \frac{\pi}{6}) \) gives \( x = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \) and \( y = 2 \times \frac{1}{2} = 1 \), resulting in Cartesian coordinates \( (\sqrt{3}, 1) \).

Mastering these conversions allows seamless navigation between forms, crucial in complex problem-solving scenarios.