Problem 55 - Precalculus with Limits

Question

In Exercises 55-64, use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. \(\left(3, -2\right)\)

Step-by-Step Solution

Answer

The polar coordinates for the point given in rectangular form (3, -2) are approximately \(\left(\sqrt{13}, \tan^{-1}\left(\frac{-2}{3}\right) +2\pi \right)\).

Step 1: Finding the magnitude of the radius

For the rectangular coordinates given, \(x = 3\) and \(y = -2\). Plug these values into the formula \(r = \sqrt{x^2 + y^2}\) to find the magnitude of the radius.

Step 2: Calculating the radius r

By plugging the values into the formula, \(r = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}\).

Step 3: Finding the angle

Use the formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\) to calculate the angle. Plug in the given coordinates \(x = 3\) and \(y = -2\).

Step 4: Calculating the angle θ

Putting the rectangular coordinates into the formula results in \(\theta = \tan^{-1}\left(\frac{-2}{3}\right)\).

Step 5: Adjusting the angle for the correct quadrant

Since the point (3, -2) is located in the fourth quadrant, an adjustment of \(\theta\) is needed. As the arctan function only returns values from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) and the point is in the fourth quadrant where θ should be more than \(\frac{\pi}{2}\), add \(2\pi\) to the calculated value of θ.

Step 6: Final polar coordinates

The polar coordinates are thus \(\left(\sqrt{13}, \tan^{-1}\left(\frac{-2}{3}\right) +2\pi \right)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to pinpoint a location on a 2-dimensional plane using two numbers. These numbers, typically known as

• "x" (horizontal axis) and
• "y" (vertical axis),

represent the precise distance of the point from the origin, which is the point (0,0).
Understanding rectangular coordinates involves plotting a point based on these two values. For example, for the point (3, -2), you would move three units to the right (along the x-axis) and two units down (along the y-axis) from the origin. Each point in rectangular coordinates can be converted into polar coordinates using the concepts of a radius and angle, involving trigonometric calculations.

Radius

The radius in polar coordinates represents the straight-line distance from the origin (the center of the coordinate system) to the point. It is analogous to the hypotenuse of a right triangle that can be formed using the point, the x-axis, and the line parallel to the y-axis through the point.
The formula to find the radius from rectangular coordinates is given by \[ r = \sqrt{x^2 + y^2} \] This formula comes from the Pythagorean theorem. By plugging the x and y values of the point (3, -2) into this formula, we calculate the radius to be \[ r = \sqrt{3^2 + (-2)^2} = \sqrt{13} \] This represents how far away the point is from the origin, regardless of its direction.

Trigonometric Functions

Trigonometric functions are mathematical tools used to relate angles to lengths. They are crucial in converting between rectangular and polar coordinates.
The tangent function, \( \tan \theta = \frac{y}{x} \), is particularly useful for finding the angle \( \theta \) formed by the radius with the positive x-axis.

• In our example, given x = 3 and y = -2, the tangent function calculates the angle using \[ \theta = \tan^{-1}\left(\frac{-2}{3}\right) \]
• The arctangent function \( \tan^{-1} \) essentially performs the inverse operation of \( \tan \theta \), providing the angle based on the y and x values.

Through these calculations, one determines the direction from the origin to the point.

Magnitude

Magnitude refers to the size or length of the radius found when converting a point from rectangular to polar coordinates. Essentially, it is the absolute distance of the point from the origin, without considering direction.

• This concept can be visualized as the length of a vector pointing straight from the origin to the point \( (x, y) \).
• For our point (3, -2), the magnitude is exactly the same as the radius, which we calculated before as \( \sqrt{13} \).

Understanding magnitude is important because it represents the scalar length of the radius needed in locating the point within the polar coordinate system.

Quadrants

The coordinate plane is divided into four sections, called quadrants. They help determine the sign and value adjustments needed during conversions between coordinate systems.

• Quadrant I is where both x and y are positive.
• Quadrant II is where x is negative and y is positive.
• Quadrant III is where both x and y are negative, and lastly, Quadrant IV is where x is positive and y is negative.

Our point (3, -2) lies in Quadrant IV.
This information is essential because it tells us about the angle's adjustment needed for polar coordinates. The arctangent function typically returns angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\),but for Quadrant IV, we need to adjust by adding \(2\pi\) to get the correct angle relative to the positive x-axis in the polar system.