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Magazine Chapter 10: Problem 3
Plot the point whose polar coordinates are given. Then ind the Cartesian coordinates of the point. (a) \((2,3 \pi / 2)\) (b) \((\sqrt{2}, \pi / 4)\) (c) \((-1,-\pi / 6)\)
Short Answer
Expert verified
(a) (0, -2); (b) (1, 1); (c) (-\frac{\sqrt{3}}{2}, \frac{1}{2}).
Step by step solution
01
Understanding Polar Coordinates
Polar coordinates are represented as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angular coordinate, measured in radians, from the positive x-axis.
02
Converting (r, 胃) to Cartesian Coordinates
The Cartesian coordinates \((x, y)\) can be calculated from polar coordinates \((r, \theta)\) using the following formulas: \(x = r \cos \theta\), \(y = r \sin \theta\).
03
Convert (2, 3蟺/2) to Cartesian Coordinates
Here, \(r = 2\) and \(\theta = \frac{3\pi}{2}\). The Cartesian coordinates are calculated as \(x = 2 \cos \frac{3\pi}{2} = 2 \times 0 = 0\) and \(y = 2 \sin \frac{3\pi}{2} = 2 \times (-1) = -2\). Thus, the point is \((0, -2)\).
04
Convert (鈭�2, 蟺/4) to Cartesian Coordinates
For \(r = \sqrt{2}\) and \(\theta = \frac{\pi}{4}\), the Cartesian coordinates are \(x = \sqrt{2} \cos \frac{\pi}{4} = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1\) and \(y = \sqrt{2} \sin \frac{\pi}{4} = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1\). Therefore, the point is \((1, 1)\).
05
Convert (-1, -蟺/6) to Cartesian Coordinates
For \(r = -1\) and \(\theta = -\frac{\pi}{6}\), the Cartesian coordinates are \(x = -1 \cos(-\frac{\pi}{6}) = -1 \times \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}\) and \(y = -1 \sin(-\frac{\pi}{6}) = -1 \times (-\frac{1}{2}) = \frac{1}{2}\). Thus, the point is \((-\frac{\sqrt{3}}{2}, \frac{1}{2})\).
06
Plotting the Points
Now, plot the resulting Cartesian coordinates on a Cartesian plane: (a) \((0, -2)\) is below the origin on the y-axis, (b) \((1, 1)\) is in the first quadrant where both x and y are positive, (c) \((-\frac{\sqrt{3}}{2}, \frac{1}{2})\) is in the second quadrant, as x is negative and y is positive.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar coordinates
Polar coordinates are a way of representing points in a plane by a distance from a designated reference point and an angle from a reference direction. This reference point is usually the origin of the plane, and the reference direction is typically the positive x-axis. The coordinates are given in the form \((r, \theta)\), where:
- \(r\) is the radial or linear distance from the origin.
- \(\theta\) is the angle measured from the positive x-axis, typically in radians.
Cartesian coordinates
Cartesian coordinates provide a simple way to describe the location of points in a plane, using a pair of numerical values known as \((x, y)\). Here鈥檚 what these represent:
- \(x\) is the horizontal distance from the origin, either to the left (negative) or right (positive).
- \(y\) is the vertical distance from the origin, either downwards (negative) or upwards (positive).
Trigonometric functions
Trigonometric functions, such as sine and cosine, are fundamental in connecting polar and Cartesian coordinates. They help us convert between the two systems with ease:
- The cosine function \(\cos\) provides the adjacent side of a right triangle formed by a point in polar coordinates for the given angle \(\theta\).
- The sine function \(\sin\) gives the opposite side for the same angle \(\theta\).
- \(r\) is the radius or distance from the origin.
- \(\theta\) is the angle in radians.
Radian measure
Radians provide a natural and straightforward measure of angles, often favored over degrees in mathematical calculations. A full circle is \(2\pi\) radians, which is equivalent to 360 degrees. Here's why radians simplify the mathematics:
- Radians are based on the arc length of a unit circle, making them dimensionless. Thus, they fit naturally in many calculus equations.
- Angle-related calculations often become simpler and more direct when expressed in radians.
