Problem 3 Write the equations that are use... [FREE SOLUTION]

Chapter 11: Problem 3

Write the equations that are used to express a point with Cartesian coordinates \((x, y)\) in polar coordinates.

Short Answer

Expert verified

Answer: The equations to convert a point in Cartesian coordinates (x, y) to polar coordinates (r, 胃) are: r = 鈭�(x虏 + y虏) 胃 = arctan2(y, x)

Step by step solution

Understand the coordinate systems

In Cartesian coordinates, each point is described by a pair of values (x, y), representing the horizontal and vertical distances from the origin. In polar coordinates, each point is described by a pair of values (r, 胃), where r is the distance from the origin and 胃 is the angle between the positive x-axis and the line connecting the origin to the point (measured counterclockwise).

Derive the relationship between x, y, and r

To find the relationship between the Cartesian and polar coordinates, think about a right triangle formed by the horizontal distance (x) and the vertical distance (y) from the origin to the point in the coordinate plane. The hypotenuse of this triangle will be the distance r from the origin to the point. By the Pythagorean theorem, we can write the relation between x, y, and r as: r^2 = x^2 + y^2 Now, solving for r, we have: r = \sqrt{x^2 + y^2}

Derive the relationship between x, y, and 胃

With the triangle described in step 2, we can now find a relationship between x, y, and 胃. By using basic trigonometric functions, we get: tan(胃) = \frac{y}{x} Since 胃 is an angle between 0 and 2蟺, we can write: 胃 = arctan\left(\frac{y}{x}\right) However, this equation doesn't account for cases where x = 0, as the tangent function would be undefined. To handle such cases, we can use the arctan2 function, which computes the arctangent of the quotient of its arguments, accounting for the correct quadrant of the result: 胃 = arctan2(y, x) This equation ensures that the angle 胃 is in the correct quadrant.

Write the final equations

After deriving the relationships, we are ready to write the equations that convert a point with Cartesian coordinates (x, y) to polar coordinates (r, 胃): r = \sqrt{x^2 + y^2} 胃 = arctan2(y, x) Now you have the equations to express a point in Cartesian coordinates as polar coordinates.

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91影视

Write the equations that are used to express a point with Cartesian coordinates \((x, y)\) in polar coordinates.

Short Answer

Step by step solution

Understand the coordinate systems

Derive the relationship between x, y, and r

Derive the relationship between x, y, and 胃

Write the final equations

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