Chapter 14: Problem 76
Find the rectangular coordinates of the point whose polar coordinates are \(\left(6, \frac{2 \pi}{3}\right)\)
Short Answer
Expert verified
The rectangular coordinates are \((-3, 3\sqrt{3})\).
Step by step solution
01
Understand Polar Coordinates
Polar coordinates \((r, \theta)\) represent a point in the plane where \(r\) is the radial distance from the origin and \( \theta \) is the angle in radians from the positive x-axis.
02
Identify the Given Coordinates
The given polar coordinates are \(\left(6, \frac{2 \pi}{3}\right)\), where \(r = 6\) and \(\theta = \frac{2 \pi}{3}\).
03
Use Conversion Formulas
To convert polar coordinates to rectangular coordinates, use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
04
Calculate Rectangular Coordinates
Substitute the polar coordinates into the formulas: \[ x = 6 \cos \left( \frac{2 \pi}{3} \right) \] \[ y = 6 \sin \left( \frac{2 \pi}{3} \right) \]
05
Find Cosine and Sine Values
Recall that \[ \cos \left( \frac{2 \pi}{3} \right) = -\frac{1}{2} \] and \[ \sin \left( \frac{2 \pi}{3} \right) = \frac{\sqrt{3}}{2} \]
06
Substitute and Simplify
Now, substitute these values into the equations: \[ x = 6 \cdot -\frac{1}{2} = -3 \] \[ y = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \]
07
Final Rectangular Coordinates
Thus, the rectangular coordinates are \((-3, 3\sqrt{3})\).
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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 using a radial distance and an angle. Instead of using the traditional x and y coordinates, polar coordinates use two values: \(r\), which is the distance from the origin to the point, and \(\theta\), which is the angle between the positive x-axis and the line connecting the origin to the point. This system is particularly useful in situations where the relationship between points is better expressed in terms of distance and direction. For example:\( r \) tells you how far the point is from the origin. \( \theta \) indicates the direction of the point relative to the x-axis.
For the given problem, the polar coordinates are \(\left(6, \frac{2 \pi}{3}\right)\). Here, 6 is the distance from the origin, and \( \frac{2 \pi}{3} \) radians is the angle.
For the given problem, the polar coordinates are \(\left(6, \frac{2 \pi}{3}\right)\). Here, 6 is the distance from the origin, and \( \frac{2 \pi}{3} \) radians is the angle.
Rectangular Coordinates
Rectangular coordinates (also called Cartesian coordinates) use the traditional x and y values to represent points in a plane. Each point is defined by its horizontal (x) and vertical (y) distances from the origin. This system is widely used and is familiar to most students from early mathematics education. It provides a straightforward way to plot points, create graphs, and perform many types of calculations in geometry and algebra. For instance:\( x \) corresponds to the horizontal axis. \( y \) corresponds to the vertical axis.
In the problem, the goal is to convert the given polar coordinates to rectangular form. We need to find the x and y values that correspond to the point \( (6, \frac{2\pi}{3}) \). We achieve this by using trigonometric functions.
In the problem, the goal is to convert the given polar coordinates to rectangular form. We need to find the x and y values that correspond to the point \( (6, \frac{2\pi}{3}) \). We achieve this by using trigonometric functions.
Trigonometry
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It provides the tools we need to convert between polar and rectangular coordinates. The key trigonometric functions involved in this conversion are sine (\(\sin\)) and cosine (\(\cos\)). These functions relate an angle in a right-angled triangle to the ratios of its sides:
In our specific case:For \( x \), we use \( x = r \cos(\theta) \). For \( y \), we use \( y = r \sin(\theta) \).
Given \( r = 6 \) and \( \theta = \frac{2 \pi}{3} \), we substitute the values:\br> \( x = 6 \cos\left( \frac{2 \pi}{3} \right) \). \( y = 6 \sin\left( \frac{2 \pi}{3} \right) \).
- \(\cos(\theta)\) gives the ratio of the adjacent side to the hypotenuse.
- \(\sin(\theta)\) gives the ratio of the opposite side to the hypotenuse.
In our specific case:
Given \( r = 6 \) and \( \theta = \frac{2 \pi}{3} \), we substitute the values:\br>
Coordinate System Conversion
Converting from polar to rectangular coordinates involves using the trigonometric relationships between the angles and distances. We apply the formulas:
\[ x = r \cos \theta \]
\[ y = r \sin \theta \]
Using the given polar coordinates \( \left( 6, \frac{2\pi}{3} \right) \), we get:\( x = 6 \cos \left( \frac{2 \pi}{3} \right) = 6 \times -\frac{1}{2} = -3 \) \( y = 6 \sin \left( \frac{2 \pi}{3} \right) = 6 \times \frac{\sqrt{3}}{2} = 3 \sqrt{3} \).
Thus, the rectangular coordinates are \( (-3, 3\sqrt{3}) \).
Remember, converting coordinates between systems enhances our ability to solve complex problems by allowing us to switch to the most convenient form.
\[ x = r \cos \theta \]
\[ y = r \sin \theta \]
Using the given polar coordinates \( \left( 6, \frac{2\pi}{3} \right) \), we get:
Thus, the rectangular coordinates are \( (-3, 3\sqrt{3}) \).
Remember, converting coordinates between systems enhances our ability to solve complex problems by allowing us to switch to the most convenient form.