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Chapter 9: Problem 41

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (-6,0) $$

Short Answer

Expert verified
The polar coordinates are \((6, \pi)\).

Step by step solution

01

Find the Radius

The radius in polar coordinates, denoted as \( r \), is the distance from the origin to the point \((-6, 0)\). This is simply the absolute value of the \( x \)-coordinate since \( y=0 \). Thus, \( r = |-6| = 6 \).
02

Determine the Angle

The angle \( \theta \) in polar coordinates is the angle formed with the positive \( x \)-axis. For the point \((-6, 0)\), which lies on the negative \( x \)-axis, \( \theta \) is \( \pi \) radians, as \( x \) is negative, and \( y \) is zero.
03

Write the Polar Coordinates

With \( r = 6 \) and \( \theta = \pi \), the polar coordinates of the point \((-6, 0)\) are \( (6, \pi) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are a way to specify a location on a plane using an ordered pair of numbers. These numbers correspond to distances along perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical). For example, the point (-6, 0) indicates that we move 6 units to the left of the origin (0, 0) on the x-axis and not move on the y-axis at all.
  • The first value in the pair, -6 in this case, tells us how far to move along the x-axis.
  • The second value, 0, tells us how far to move up or down the y-axis.
By using these two values together, we can accurately plot any point on the plane.
Angle Conversion
Converting from rectangular to polar coordinates involves determining an angle, 胃, which is the angle formed with the positive x-axis.
  • If a point falls on the negative x-axis, like (-6, 0), the angle must be carefully calculated.
  • Angles on the negative x-axis are directly opposite those on the positive side, meaning they measure 蟺 radians.
This ensures that the angle accurately represents the direction of the point from the origin.
Radius Calculation
The radius, represented by r, is a crucial part of polar coordinates. It shows the distance from the origin to our point on the plane. For points on the x-axis, the radius is just the absolute value of the x-coordinate.
  • In our example, the point (-6, 0), we find the radius by calculating | -6 | = 6.
  • This distance remains positive as it represents the magnitude of the position from the center.
Thus, the radius here tells us that despite the negative direction, the point is 6 units away from the origin.
Radians
Radians are a way of measuring angles, where the angle is described in terms of the radius of a circle. In mathematics, radians are often used over degrees because they allow for more straightforward integration into mathematical formulas and physics.
  • One complete revolution around a circle is 2蟺 radians.
  • A straight angle, or half a circle, is equated to 蟺 radians, as seen in our calculation for the point on the negative x-axis.
By expressing the angle as 蟺 radians, we adhere to the standard conventions used in many mathematical calculations beyond just polar coordinates.

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Most popular questions from this chapter

Show that the graph of \(r=a \cos \theta+b \sin \theta\) is a circle, and find its center and radius.

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (3, \pi / 2) $$

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (6 \sqrt{2}, 11 \pi / 6) $$

(a) Let \(w=\cos \frac{2 \pi}{n}+i \sin \frac{2 \pi}{n}\) where \(n\) is a positive integer. Show that \(1, w, w^{2}, w^{3}, \ldots, w^{n-1}\) are the \(n\) distinct nth roots of 1 . (b) If \(z \neq 0\) is any complex number and \(s^{n}=z,\) show that the \(n\) distinct \(n\) th roots of \(z\) are $$ s, s w, s w^{2}, s w^{3}, \ldots, s w^{n-1} $$

Convert the polar equation to rectangular coordinates. $$ r=6 \cos \theta $$
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