Chapter 15: Problem 36
Write the rectangular coordinates of each point. $$\left(18.3, \frac{2 \pi}{3}\right)$$
Short Answer
Expert verified
The rectangular coordinates are approximately \(x = -9.15, y = 15.8796\).
Step by step solution
01
Understanding the Polar Coordinates
The given point \(18.3, \frac{2 \pi}{3}\) is in polar coordinates, where the first number represents the radius \(r\) and the second number represents the angle \(\theta\) in radians.
02
Converting to Rectangular Coordinates
To convert from polar to rectangular coordinates, use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
03
Calculate the X-Coordinate
Calculate the x-coordinate using the cosine function: \( x = 18.3 \cos\left(\frac{2 \pi}{3}\right)\). Since \cos\left(\frac{2 \pi}{3}\right) = -\frac{1}{2}, x = 18.3 \times -\frac{1}{2} = -9.15.
04
Calculate the Y-Coordinate
Calculate the y-coordinate using the sine function: \(y = 18.3 \sin\left(\frac{2 \pi}{3}\right)\). Since \sin\left(\frac{2 \pi}{3}\right) = \frac{\sqrt{3}}{2}, y = 18.3 \times \frac{\sqrt{3}}{2} = 15.8796.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Coordinates
Polar coordinates provide a method of representing points in a plane using a radius and an angle. In this system, a point's location is determined by how far away it is from a central point (the origin), and the angle it makes with a reference direction, usually the positive x-axis.
The system is especially useful in situations where the relationship between two points is easier to model with angles and distances. Examples include the orbits of celestial bodies and the behavior of waves. Each point is written as a pair \( (r, \theta) \) where \( r \) is the radius or distance from the origin, and \( \theta \) is the angle in radians or degrees from the reference direction.
Understanding polar coordinates paves the way for important applications in fields such as physics, engineering, and navigation.
The system is especially useful in situations where the relationship between two points is easier to model with angles and distances. Examples include the orbits of celestial bodies and the behavior of waves. Each point is written as a pair \( (r, \theta) \) where \( r \) is the radius or distance from the origin, and \( \theta \) is the angle in radians or degrees from the reference direction.
Understanding polar coordinates paves the way for important applications in fields such as physics, engineering, and navigation.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, allow for the representation of points with a two-dimensional grid system. This system utilizes two perpendicular axes: the horizontal axis, known as the x-axis, and the vertical axis called the y-axis.
Points in this plane are identified by order pairs \( (x, y) \), where \( x \) is the distance along the x-axis, and \( y \) is the distance along the y-axis. Rectangular coordinates are incredibly prevalent in mathematics and serve as the foundation of algebra and calculus.
Being well-versed in this system is crucial for technical tasks, such as computer graphics programming, structural design, and much more. The ease of plotting functions and the straightforward distance calculations make rectangular coordinates an indispensable tool in mathematics.
Points in this plane are identified by order pairs \( (x, y) \), where \( x \) is the distance along the x-axis, and \( y \) is the distance along the y-axis. Rectangular coordinates are incredibly prevalent in mathematics and serve as the foundation of algebra and calculus.
Being well-versed in this system is crucial for technical tasks, such as computer graphics programming, structural design, and much more. The ease of plotting functions and the straightforward distance calculations make rectangular coordinates an indispensable tool in mathematics.
Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between side lengths and angles of triangles. Its principles are broadly used to convert polar coordinates to rectangular coordinates.
Using the trigonometric functions sine (sin) and cosine (cos), we can calculate the x and y coordinates in a rectangular coordinate system based on a given radius and angle from polar coordinates.
For any right-angled triangle, if \( \theta \) is one of the non-right angles, and \( r \) is the hypotenuse, then the side adjacent to the angle is \( r \cdot \cos(\theta) \) and the side opposite the angle is \( r \cdot \sin(\theta) \). Trigonometry is not just a part of academic curricula; it's extensively used in real-world applications such as engineering, physics, and even in determining the height of tides in oceans.
Using the trigonometric functions sine (sin) and cosine (cos), we can calculate the x and y coordinates in a rectangular coordinate system based on a given radius and angle from polar coordinates.
For any right-angled triangle, if \( \theta \) is one of the non-right angles, and \( r \) is the hypotenuse, then the side adjacent to the angle is \( r \cdot \cos(\theta) \) and the side opposite the angle is \( r \cdot \sin(\theta) \). Trigonometry is not just a part of academic curricula; it's extensively used in real-world applications such as engineering, physics, and even in determining the height of tides in oceans.
Radian Measure
Radians provide an alternative to degrees for measuring angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. In a full circle, there are \( 2\pi \) radians, which is equivalent to 360 degrees.
Radian measure is commonly used in higher mathematics because it simplifies many formulas, and it's the standard unit of angular measure in mathematical analysis involving trigonometry functions. In our example, when we see the angle \( \frac{2 \pi}{3} \) radians, this represents one-third of a full circle. Understanding how to work with radians is important when using trigonometric functions, as they often require angles to be in radians for accurate calculations.
Radian measure is commonly used in higher mathematics because it simplifies many formulas, and it's the standard unit of angular measure in mathematical analysis involving trigonometry functions. In our example, when we see the angle \( \frac{2 \pi}{3} \) radians, this represents one-third of a full circle. Understanding how to work with radians is important when using trigonometric functions, as they often require angles to be in radians for accurate calculations.