Problem 33
Question
In Exercises 29-36, use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. \(\left(2.5, 1.58\right)\)
Step-by-Step Solution
Verified Answer
The rectangular coordinates of the point in polar coordinates \((2.5, 1.58)\) rounded to two decimal places are \((x, y)\)
1Step 1: Identify the given polar coordinates
The given polar coordinates are \(r = 2.5\) and \(\theta = 1.58\).
2Step 2: Convert the angle from degrees to radians
The given angle is in radians already, so no conversion is required.
3Step 3: Find the x-coordinate
To find the x-coordinate, plug the given values into the formula \(x = r \cos(\theta)\). Therefore, \(x = 2.5 \cos(1.58)\). Use a graphing utility or scientific calculator to find the value, and make sure to round your results to two decimal places.
4Step 4: Find the y-coordinate
To find the y-coordinate, again plug the given values into the formula \(y = r \sin(\theta)\). Therefore, \(y = 2.5 \sin(1.58)\). Use a graphing utility or scientific calculator to find the value, and again remember to round your results to two decimal places.
Key Concepts
Polar CoordinatesRectangular CoordinatesGraphing UtilityTrigonometric Functions
Polar Coordinates
Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is known as the pole, and is similar to the origin in the rectangular (Cartesian) coordinate system. The reference direction, usually the positive x-axis, is called the polar axis.
In the form \( (r, \theta) \), \(r\) represents the radial distance from the pole, while \(\theta\) is the angle measured in radians from the polar axis. To visualize this, imagine a line from the pole to the point: \(r\) is the length of this line, and \(\theta\) is the angle it makes with the polar axis.
Polar coordinates can be particularly useful in situations where the relationship between two points is easier to express with angles and distances (e.g., in trigonometry, navigation, and physics), rather than Cartesian coordinates.
In the form \( (r, \theta) \), \(r\) represents the radial distance from the pole, while \(\theta\) is the angle measured in radians from the polar axis. To visualize this, imagine a line from the pole to the point: \(r\) is the length of this line, and \(\theta\) is the angle it makes with the polar axis.
Polar coordinates can be particularly useful in situations where the relationship between two points is easier to express with angles and distances (e.g., in trigonometry, navigation, and physics), rather than Cartesian coordinates.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, define a point on a plane using two perpendicular lines, one horizontal (the x-axis) and one vertical (the y-axis), which intersect at a point called the origin.
A point's location in this system is indicated by an ordered pair of numbers, \( (x, y) \), where \(x\) is the distance from the y-axis, and \(y\) is the distance from the x-axis. The horizontal value, or \(x\), can be thought of as how far 'left' or 'right' the point is, while the vertical value, or \(y\), is how far 'up' or 'down' it is.
The primary advantage of rectangular coordinates over other coordinate systems is the ease of plotting points and performing algebraic computations. They are particularly helpful in graphing equations involving polynomials, and in most geometric and algebra calculations.
A point's location in this system is indicated by an ordered pair of numbers, \( (x, y) \), where \(x\) is the distance from the y-axis, and \(y\) is the distance from the x-axis. The horizontal value, or \(x\), can be thought of as how far 'left' or 'right' the point is, while the vertical value, or \(y\), is how far 'up' or 'down' it is.
The primary advantage of rectangular coordinates over other coordinate systems is the ease of plotting points and performing algebraic computations. They are particularly helpful in graphing equations involving polynomials, and in most geometric and algebra calculations.
Graphing Utility
A graphing utility, such as a graphing calculator or software like Desmos, Geogebra, or Matlab, is an invaluable tool for students. It helps in visualizing mathematical concepts, solving equations, and checking results.
In the context of polar and rectangular coordinates, graphing utilities simplify the process of conversion and plotting. They can produce the graph of a function by computing the coordinates in real-time and can easily switch between different coordinate systems. Most importantly, they make it easy to work with trigonometric functions, offering both numerical and graphical insights into their behavior.
In the context of polar and rectangular coordinates, graphing utilities simplify the process of conversion and plotting. They can produce the graph of a function by computing the coordinates in real-time and can easily switch between different coordinate systems. Most importantly, they make it easy to work with trigonometric functions, offering both numerical and graphical insights into their behavior.
Trigonometric Functions
x = r cos(θ)
), while the sine function gives us the \(y\)-coordinate (y = r sin(θ)
). These functions are periodic, meaning they repeat values in a predictable pattern, which is essential when analyzing waves or circular motion in various fields of science and engineering.Relevance in Coordinate Conversion
Understanding sine and cosine is critical when working with polar coordinates. Since the angle \(\theta\) dictates the direction of the radial line \(r\) from the pole, the trigonometric functions sine and cosine determine the vertical and horizontal displacement of this line, leading us to the corresponding \(x\) and \(y\) values in rectangular coordinates.Other exercises in this chapter
Problem 33
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