Problem 61 - Precalculus

Question

Use a graphing utility to find one set of polar coordinates of the point given in rectangular coordinates. $$(3,-2)$$

Step-by-Step Solution

Answer

The given point (3,-2) converted from rectangular to polar coordinates is approximately \((\sqrt{13}, 5.30)\).

Step 1: Identify the rectangular coordinates

For the point \((3,-2)\), the x-coordinate is 3 and the y-coordinate is -2.

Step 2: Calculate the r value

We first find the value of r using the formula \(r = \sqrt{x^2 + y^2}\). Substituting x as 3 and y as -2 into the formula, we get \(r = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}\). So, r equals \(\sqrt{13}\).

Step 3: Calculate \(\Theta\) value

We then compute the value of \(\Theta\) via the formula \(\Theta = \arctan(\frac{y}{x})\). Bearing in mind that our y-coordinate is negative and our x-coordinate is positive, indicating that the point is in the fourth quadrant, we use the adjusted formula for the fourth quadrant, \(\Theta = \arctan(\frac{y}{x}) + 2\pi\). Substituting x as 3 and y as -2 into the formula, we get \(\Theta = \arctan(\frac{-2}{3}) + 2\pi\). Computing this using a calculator, we get \(\Theta \approx 5.30\) (in radians).

Step 4: Assemble the polar coordinates

Finally, we put together the r value and the \(\Theta\) value into a pair of polar coordinates, \((r, \Theta)\). Thus, the point in polar coordinates is \((\sqrt{13}, 5.30)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, consist of two values: the x-coordinate (horizontal position) and the y-coordinate (vertical position). For example, in the point (3, -2), 3 is the x-coordinate, and -2 is the y-coordinate. These coordinates are incredibly useful for plotting points on a standard graph.

They help us visualize where exactly a point lies on a grid. To convert these into polar coordinates, we need to understand both the distance from the origin and the angle from the positive x-axis.

Quadrants

In the Cartesian plane, quadrants help us understand the position of a point relative to the axes. The plane is divided into four quadrants:

• Quadrant I: x and y are both positive.
• Quadrant II: x is negative and y is positive.
• Quadrant III: x and y are both negative.
• Quadrant IV: x is positive and y is negative.

For the point (3, -2), since the x-coordinate is positive and the y-coordinate is negative, it falls in Quadrant IV. This information is crucial for determining the correct angle when converting to polar coordinates.

Arctan Formula

The arctan formula is used to find angles in the conversion from rectangular to polar coordinates. The formula \(\Theta = \arctan(\frac{y}{x})\) gives the basic angle for the point relative to the x-axis. However, in different quadrants, adjustments must be made to ensure accuracy:

• Quadrant I: Use \(\Theta = \arctan(\frac{y}{x})\).
• Quadrant IV: Use \(\Theta = \arctan(\frac{y}{x}) + 2\pi\) to adjust for the full rotation.

For a point in Quadrant IV like (3, -2), compute the arctan and then adjust by adding \(2\pi\) to convert the angle to its proper measure in radians.

Radians

Radians are a way to measure angles, used commonly in mathematics because they relate angle measures directly to arc length in a circle. One full circle (360 degrees) is equivalent to \(2\pi\) radians. This unit is preferred in many mathematical computations due to its natural properties.

When converting rectangular coordinates like (3, -2) into polar form, we often express the angle in radians. This makes further calculations, especially involving trigonometric functions, more straightforward.