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Chapter 6: Problem 37

A point in rectangular coordinates is given. Convert the point to polar coordinates. (1,1)

Short Answer

Expert verified
The polar coordinates for the point (1,1) in the rectangular coordinate system are \( (\sqrt{2}, \pi/4) \)

Step by step solution

01

Calculation of distance r

The distance r from the origin to the point (x,y) can be calculated using the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \). Substituting the given x=1 and y=1 into the equation, we get \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \).
02

Calculation of angle 胃

The angle 胃 from the positive x-axis to the line segment connecting the origin to the point can be found using the tangent formula: \( 胃 = \arctan{(y/x)} \). Substituting the given x=1 and y=1 into the equation, we get \( 胃 = \arctan{(1/1)} = \arctan{1} \) which is \( \pi/4 \) in radians or 45 degrees in degrees.
03

Presenting the final polar coordinates

The polar coordinates are usually written in the form (r, 胃). So, the polar coordinates for the point (1,1) are \( (\sqrt{2}, \pi/4) \)

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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, represent a point in a plane using two values: x and y.
These coordinates provide a straightforward way to specify the position of a point by its distance along two perpendicular axes.
  • The x-coordinate indicates how far left or right a point is from the origin (0,0).
  • The y-coordinate specifies how far up or down the point is from the origin.
For example, the point (1,1) tells us to move 1 unit to the right and 1 unit up from the origin. This system is not only intuitive but also very useful for performing calculations related to distance and angles.
Conversion
Converting between rectangular coordinates and other coordinate systems, like polar coordinates, involves a transformation of how we describe a point. To convert a point from rectangular (x, y) coordinates to polar coordinates (r, 胃), we follow two main steps:
  • Find the distance r: This is the distance from the origin (0,0) to the point (x, y). We use the Pythagorean theorem, which we'll discuss shortly, to find this.
  • Determine the angle 胃: This angle is formed between the positive x-axis and the line connecting the origin to the point. For this, we use trigonometric functions.
These conversions are vital for various fields, including physics and engineering, where rotational motion or radial distances are important.
Pythagorean Theorem
The Pythagorean theorem is a fundamental concept in geometry and is crucial for converting rectangular coordinates to polar coordinates. It states that in a right-angled triangle, the square of the length of the hypotenuse (r) is equal to the sum of the squares of the lengths of the other two sides (x and y).
Here's the formula:
  • \( r = \sqrt{x^2 + y^2} \)
By substituting the point's coordinates, \( (1,1) \), we calculate the distance from the origin to this point. This calculation shows how the Pythagorean theorem provides us with the radial component, r, for polar coordinates.Understanding this gives you a clearer view of the geometric link between different types of coordinates.
Trigonometric Functions
Trigonometric functions help us understand the relationship between the angles and the sides of a triangle. In the context of converting coordinates, they help us find the angle 胃 in polar coordinates. The tangent function, abbreviated as tan, is especially helpful for finding 胃.
  • Formula: \( \theta = \arctan{(y/x)} \)
In our example with the point (1,1), the tangent of the angle the point makes with the x-axis is 1 (since \( 1/1 \)). The arctangent, or inverse tangent, gives us \( \pi/4 \) radians or 45 degrees. These trigonometric calculations are essential for understanding points in a polar system, where angles are a key component.

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