Problem 22 - Calculus Early Transcendentals

Question

Express the following Cartesian coordinates in polar coordinates in at least two different ways. (-1,0)

Step-by-Step Solution

Answer

Question: Convert the Cartesian coordinates (-1,0) into polar coordinates in two different ways. Answer: The polar coordinates are (1, \(180^\circ\)) or (1, \(\pi\)) and (1, \(540^\circ\)) or (1, \(3\pi\)).

Step 1: Identify the Cartesian coordinates

(-1, 0) is our given Cartesian coordinate, where x = -1 and y = 0.

Step 2: Convert into polar coordinate (First way)

To do this, we need to find the magnitude (r) and the angle (θ): Calculate the magnitude (\(r\)): \(r = \sqrt{x^2 + y^2}\) \(r = \sqrt{(-1)^2 + 0^2}\) \(r = \sqrt{1}\) \(r = 1\) Calculate the angle (θ): Since y = 0 and x = -1, θ can be directly calculated as it lies on a negative x-axis. θ = 180° or \(θ = \pi\) radians So, the polar coordinate is (1, \(180^\circ\)) or (1, \(\pi\))-(depending on whether you represent the angle in degrees or radians).

Step 3: Convert into polar coordinate (Second way)

To find another representation of the polar coordinate, we can add/subtract a multiple of 360° or 2 \(\pi\) radians to the angle found in Step 2 because this will bring us to the same point on the plane. Add 360° to the original angle: θ = 180° + 360° θ = 540° Or using radians: θ = \(\pi\) + 2\(\pi\) θ = 3\(\pi\) So, the second representation of the polar coordinate is (1, \(540^\circ\)) or (1, \(3\pi\)).

Key Concepts

Cartesian Coordinates

In the world of mathematics, Cartesian coordinates are a way to precisely locate points within a two-dimensional plane. This coordinate system is named after the French philosopher and mathematician René Descartes. Points in this system are defined using a pair of numbers, \(x,y\), where \x\ represents the horizontal position and \y\ the vertical position.

For example, the Cartesian coordinates \(-1, 0\) specify a point that is 1 unit to the left of the origin along the x-axis, and has no movement along the y-axis, leaving it directly on the x-axis. Cartesian coordinates are incredibly useful because they make it easy to graph different equations, analyze geometric figures, and connect algebra with geometry. Plus, they provide a straightforward way to convert these points into polar coordinates, allowing us to explore relationships in another dimension.

Magnitude

Magnitude in the context of polar coordinates represents the distance of a point from the origin in the coordinate plane. It's similar to how we think of the length of a vector. To calculate this magnitude, denoted by \(r\), we use the Pythagorean theorem. The formula is: \[r = \sqrt{x^2 + y^2}\]

Starting with our given Cartesian coordinates of \(-1, 0\), the calculated magnitude is:

• Substituting \x = -1\ and \y = 0\ into the formula: \[r = \sqrt{(-1)^2 + 0^2} = \sqrt{1} = 1\]

This tells us that the point is 1 unit away from the origin, confirming its position on the x-axis. Magnitude is crucial in converting Cartesian coordinates to polar form, as it determines half of the polar expression, while the other half is given by the angle.

Angle Conversion

Angle conversion is a key step in transforming Cartesian coordinates into polar coordinates. The angle, denoted as \(\theta\), indicates the direction of the point relative to the positive x-axis. This can be measured either in degrees or in radians. For our point \(-1, 0\), understanding the angle is straightforward. It lies on the negative x-axis.

Thus, the angle is:

• 180° in degrees – the point is on the opposite end of the x-axis from 0°.
• \(\pi\) radians – since radians measure the angle around the circle, and \(\pi\) signifies a straight line halfway around the unit circle.

To find an alternative polar representation, we may add or subtract full rotations, given by 360° or \2\pi\ radians. This will not change the point's position because every full rotation leads to the same angle. Hence, additional representations of these coordinates are 540° (or \3\pi\ radians), which maintain the same end location.