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Magazine Chapter 10: Problem 4
Plot the points whose polar coordinates are \(\left(3, \frac{9}{4} \pi\right)\), \(\left(-2, \frac{1}{2} \pi\right), \quad\left(-2,-\frac{1}{3} \pi\right), \quad(-1,-1), \quad(1,-7 \pi), \quad\left(-3,-\frac{1}{6} \pi\right)\), \(\left(-2,-\frac{1}{2} \pi\right)\), and \(\left(3,-\frac{33}{2} \pi\right) .\)
Short Answer
Expert verified
Convert polar to Cartesian coordinates to plot each point.
Step by step solution
01
Understand Polar Coordinates
Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured in radians from the positive x-axis. Positive \(r\) indicates a distance in the direction of the angle, while negative \(r\) means you have to go in the opposite direction of the angle.
02
Convert to Cartesian Coordinates
To plot the points, it's often helpful to convert polar coordinates to Cartesian coordinates (x, y) using the formulas:\[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]
03
Plot Point \((3, \frac{9}{4}\pi)\)
First, calculate the Cartesian coordinates:\[ x = 3 \cos\left(\frac{9}{4}\pi\right) \]\[ y = 3 \sin\left(\frac{9}{4}\pi\right) \]An angle of \(\frac{9}{4}\pi\) can be simplified to \(\frac{5}{4}\pi\), which places the angle in the third quadrant, resulting in negative x and y coordinates.
04
Plot Point \((-2, \frac{1}{2}\pi)\)
Here, calculate:\[ x = -2 \cos\left(\frac{1}{2}\pi\right) = 0 \]\[ y = -2 \sin\left(\frac{1}{2}\pi\right) = -2 \]This places the point on the negative y-axis.
05
Plot Point \((-2,-\frac{1}{3}\pi)\)
Calculate:\[ x = -2 \cos\left(-\frac{1}{3}\pi\right) = -2 \times \frac{1}{2} = -1 \]\[ y = -2 \sin\left(-\frac{1}{3}\pi\right) = -2 \times \left(-\frac{\sqrt{3}}{2}\right) = \sqrt{3} \]This places the point in the second quadrant.
06
Plot Point \((-1,-1)\)
Convert by calculating:\[ x = -1 \cos(-1) \]\[ y = -1 \sin(-1) \]Because the angle \(-1\) is in radians, this calculation places the point close to the origin near the positive x-axis.
07
Plot Point \((1,-7\pi)\)
Simplify the angle:\(-7\pi = -3.5\pi = -\pi/2\)\[ x = 1 \cos(-\pi/2) = 0 \]\[ y = 1 \sin(-\pi/2) = -1 \]This places the point on the negative y-axis.
08
Plot Point \((-3,-\frac{1}{6}\pi)\)
Calculate:\[ x = -3 \cos(-\frac{1}{6}\pi) = -\frac{3\sqrt{3}}{2} \]\[ y = -3 \sin(-\frac{1}{6}\pi) = \frac{3}{2} \]This places the point in the second quadrant.
09
Plot Point \((-2,-\frac{1}{2}\pi)\)
As before in Step 4, this point with the same coordinates lies on the positive y-axis:\[ x = 0, \; y = +2 \]
10
Plot Point \((3,-\frac{33}{2}\pi)\)
Simplify angle:\(-\frac{33}{2}\pi = -16.5\pi = -\pi/2\)\[ x = 3 \cos(-\pi/2) = 0 \]\[ y = 3 \sin(-\pi/2) = -3 \]This with simplification places the point on the negative y-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cartesian coordinates
The Cartesian coordinate system is the most common system for representing points on a plane. It uses two perpendicular lines, typically called the x-axis (horizontal) and y-axis (vertical), to identify a point by its distance from each axis. These distances are known as the x-coordinate (horizontal displacement) and y-coordinate (vertical displacement).
- x-coordinate: Represents how far along the point is on the horizontal axis. Positive values are to the right and negative values to the left.
- y-coordinate: Represents the vertical position. Positive values are up the y-axis, and negative ones are down.
radians
Radians are a unit of measure for angles used frequently in mathematics instead of degrees. They have a direct relationship with the radius of a circle. One full circle is equal to \( 2\pi \) radians.
- Why radians? Radians simplify many mathematical computations, especially those involved with trigonometric functions because they relate directly to the arc length of the circle.
- Conversion to degrees: Since \( 2\pi \) radians equal \( 360^{\circ}\), you can convert radians to degrees by multiplying the radian measure by \( \frac{180}{\pi} \).
coordinate transformation
Coordinate transformation involves switching between different coordinate systems, most commonly from polar to Cartesian coordinates. This operation is essential when dealing with geometrical problems where the orientation and position of points are better understood in another system.
In polar coordinates, every point is defined by a distance (\( r \)) from the origin and an angle (\( \theta \)) from the positive x-axis.
In polar coordinates, every point is defined by a distance (\( r \)) from the origin and an angle (\( \theta \)) from the positive x-axis.
- Transformation formulas: To convert from polar \( (r, \theta) \) to Cartesian \( (x, y) \), use the formulas:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- Positive and Negative r: A negative radial distance \( r \) indicates that the point lies in the opposite direction of the angle \( \theta \).
plotting points
Plotting points involves accurately placing them on a grid based on their coordinates, which can be either polar or Cartesian. This activity often requires a transformation if given in polar form.
Here鈥檚 how you go about placing points:
Here鈥檚 how you go about placing points:
- Start with transformation: If starting with polar coordinates, convert them into Cartesian coordinates using the equations for \( x \) and \( y \) to identify exact locations on the plane.
- Placement on the grid: Once coordinates are in \( (x, y) \) form, position each point by moving horizontally by the x-coordinate and vertically by the y-coordinate.
- Check for quadrant: Ensure that the sign of your coordinates guides placement into the correct quadrant:
- Positive \( x, y \) places the point in the first quadrant.
- Negative \( x, \) positive \( y \) goes to the second quadrant.
- Negative \( x, y \) goes to the third quadrant.
- Positive \( x, \) negative \( y \) leads to the fourth quadrant.
