Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the points whose polar coordinates are \((3,2 \pi)\), \(\left(2, \frac{1}{2} \pi\right),\left(4,-\frac{1}{3} \pi\right),(0,0),(1,54 \pi),\left(3,-\frac{1}{6} \pi\right),\left(1, \frac{1}{2} \pi\right)\), and \(\left(3,-\frac{3}{2} \pi\right) .\)

Step-by-Step Solution

Answer

Convert polar coordinates to Cartesian and plot: (3,0), (0,2), (2√3,-2), (0,0), (1,0), (3√3/2,-3/2), (0,1), (0,-3).

Step 1: Understand Polar Coordinates

Polar coordinates are defined by a distance from the origin \(r\) and an angle \(\theta\) from the positive x-axis. Here, \(r\) is the radial coordinate, and \(\theta\) is the angular coordinate measured in radians.

Step 2: Convert Polar to Cartesian Coordinates

Use the formulas: \(x = r \cdot \cos(\theta)\)\(y = r \cdot \sin(\theta)\)For each point, compute the Cartesian coordinates to easily plot them on the Cartesian plane.

Step 3: Point (3, 2π)

For \((3, 2\pi)\):- \(x = 3 \cdot \cos(2\pi) = 3 \cdot 1 = 3\)- \(y = 3 \cdot \sin(2\pi) = 3 \cdot 0 = 0\)Point: \((3, 0)\)

Step 4: Point (2, 1/2π)

For \(\left(2, \frac{1}{2}\pi\right)\):- \(x = 2 \cdot \cos\left(\frac{1}{2}\pi\right) = 0\)- \(y = 2 \cdot \sin\left(\frac{1}{2}\pi\right) = 2\)Point: \((0, 2)\)

Step 5: Point (4, -1/3π)

For \(\left(4,-\frac{1}{3}\pi\right)\):- \(x = 4 \cdot \cos\left(-\frac{1}{3}\pi\right) = 2\sqrt{3}\)- \(y = 4 \cdot \sin\left(-\frac{1}{3}\pi\right) = -2\)Point: \((2\sqrt{3}, -2)\)

Step 6: Point (0, 0)

For \((0, 0)\):- This point remains the origin, \((0, 0)\).

Step 7: Point (1, 54π)

For \((1, 54\pi)\):- \(x = 1 \cdot \cos(54\pi) = 1\)- \(y = 1 \cdot \sin(54\pi) = 0\)Point: \((1, 0)\)

Step 8: Point (3, -1/6π)

For \(\left(3,-\frac{1}{6}\pi\right)\):- \(x = 3 \cdot \cos\left(-\frac{1}{6}\pi\right) = \frac{3\sqrt{3}}{2}\)- \(y = 3 \cdot \sin\left(-\frac{1}{6}\pi\right) = -\frac{3}{2}\)Point: \(\left(\frac{3\sqrt{3}}{2}, -\frac{3}{2}\right)\)

Step 9: Point (1, 1/2π)

For \(\left(1, \frac{1}{2}\pi\right)\):- \(x = 1 \cdot \cos\left(\frac{1}{2}\pi\right) = 0\)- \(y = 1 \cdot \sin\left(\frac{1}{2}\pi\right) = 1\)Point: \((0, 1)\)

Step 10: Point (3, -3/2π)

For \(\left(3,-\frac{3}{2}\pi\right)\):- \(x = 3 \cdot \cos\left(-\frac{3}{2}\pi\right) = 0\)- \(y = 3 \cdot \sin\left(-\frac{3}{2}\pi\right) = -3\)Point: \((0, -3)\)

Step 11: Plot Points on Cartesian Plane

Using the converted Cartesian coordinates, you can now plot the points:- \((3, 0)\)- \((0, 2)\)- \((2\sqrt{3}, -2)\)- \((0, 0)\)- \((1, 0)\)- \(\left(\frac{3\sqrt{3}}{2}, -\frac{3}{2}\right)\)- \((0, 1)\)- \((0, -3)\)Locate these on the x-y plane with correct scaling.

Key Concepts

Coordinate Conversion

Polar coordinates are a way to define a point in a plane using two numbers:

• The radial coordinate \(r\) which is the distance from the point to the origin.
• The angular coordinate \(\theta\) which is the angle measured from the positive x-axis in a counter-clockwise direction.

To convert these polar coordinates to Cartesian coordinates, which use \((x,y)\) format, we use the following formulas:

• \(x = r \cdot \cos(\theta)\)
• \(y = r \cdot \sin(\theta)\)

These conversions allow you to translate the radial distance and angle into an (x, y) coordinate pair, making it easier to plot and understand the location and layout of points in a plane.

Cartesian Coordinates

The Cartesian coordinate system is a method of defining a point's location in a plane using two perpendicular lines, typically called the x-axis and y-axis. Each point in this system is represented by an \((x, y)\) coordinate.
The x-value indicates how far to move left or right and the y-value indicates how far to move up or down from the origin, which is the point where the x-axis and y-axis intersect.

• A positive x-value moves the point to the right of the origin, while a negative x-value moves it to the left.
• A positive y-value moves the point upward, while a negative y-value moves it downward.

Understanding Cartesian coordinates is essential when plotting points converted from polar coordinates, as it provides a straightforward method to visualize and navigate the plane.

Radian Measurement

Radians are another way to measure angles, alongside degrees. Unlike degrees, which split a circle into 360 equal parts, radians are based on the radius of the circle.
In a complete circle, there are \(2\pi\) radians, an important value to recognize:

• 1 full circle = \(360^\circ = 2\pi\) radians
• \(\pi\) radians represents a half circle or \(180^\circ\).

When dealing in polar coordinates, angles are often expressed in radians. Using radians can simplify calculations, especially when converting to Cartesian coordinates, because of their direct relationship with the \(\pi\) constant.
While it might take some time to get used to interpreting radians visually or mentally, it is a fundamental concept in trigonometry and calculus.

Plotting Points

Once you have converted polar coordinates into Cartesian coordinates, plotting them becomes a simpler task. Here's how to effectively plot points on a Cartesian plane:
1. **Define the Axes:** Make sure you have an x-axis and y-axis clearly marked. They should intersect at the origin \((0, 0)\).
2. **Locate the Points:** Use the Cartesian coordinates \((x, y)\) to find each point. Start at the origin; move right or left for the x-coordinate, then move up or down for the y-coordinate.
3. **Mark the Point:** Once you reach the correct position, mark it on the plane.
Each point gives a spatial perspective of where it lies in relation to other points. This is particularly useful when analyzing patterns or relationships between different data points.