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Chapter 7: Problem 28

Convert the polar coordinates of each point to rectangular coordinates. $$(2, \pi / 4)$$

Short Answer

Expert verified
(\sqrt{2}, \sqrt{2})

Step by step solution

01

Understand the Polar Coordinates

Polar coordinates are given as \(r, \theta\). Here, \(r = 2\) is the radius or distance from the origin, and \(\theta = \frac{\pi}{4}\) is the angle measured counterclockwise from the positive x-axis.
02

Recall the Conversion Formulas

To convert from polar coordinates \(r, \theta\) to rectangular coordinates \(x, y\), use the formulas: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]
03

Compute the Rectangular Coordinates

Apply the conversion formulas using \(r = 2\) and \(\theta = \frac{\pi}{4}\): \[ x = 2 \cos\left(\frac{\pi}{4}\right) = 2 \left(\frac{\sqrt{2}}{2}\right) = \sqrt{2}\] \[ y = 2 \sin\left(\frac{\pi}{4}\right) = 2 \left(\frac{\sqrt{2}}{2}\right) = \sqrt{2}\]
04

Write the Final Rectangular Coordinates

The rectangular coordinates corresponding to the polar coordinates \(2, \frac{\pi}{4}\) are \((\sqrt{2}, \sqrt{2})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates represent points in a plane using a distance and an angle. The distance, denoted as 饾憻, is the length from the origin to the point. The angle, denoted as 饾渻, is measured counterclockwise from the positive x-axis. In simpler terms, if you imagine standing at the origin, 饾憻 tells you how far to walk, and 饾渻 tells you which direction to go.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, use perpendicular axes to determine a point in a plane. They consist of an x-coordinate and a y-coordinate. The x-coordinate tells you how far to move horizontally from the origin, and the y-coordinate tells you how far to move vertically. Combining these two movements gives you the exact position of the point.
Trigonometric Functions
Trigonometric functions like sine and cosine are essential for converting polar coordinates to rectangular coordinates. For any angle 饾渻:
  • The cosine function, written as \(\text{cos}(饾渻)\), gives the ratio of the adjacent side to the hypotenuse in a right triangle.
  • The sine function, written as \(\text{sin}(饾渻)\), gives the ratio of the opposite side to the hypotenuse.
These functions help translate the distance and angle from polar coordinates into horizontal and vertical distances (x and y).
Coordinate Geometry
Coordinate geometry combines algebra and geometry to study points, lines, and shapes in a plane. In this context, it involves switching between different coordinate systems. Converting from polar to rectangular coordinates involves:
  • Understanding both coordinate systems.
  • Using trigonometric functions for conversion.
By applying the formulas \( x = r \text{cos}(饾渻) \) and \( y = r \text{sin}(饾渻) \), you can find the exact rectangular coordinates for given polar coordinates.

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