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Chapter 6: Problem 6

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(7, \frac{\pi}{4}\right)\)

Short Answer

Expert verified
The rectangular coordinates of the given point in polar coordinates \(\left(7, \frac{\pi}{4}\right)\) are: \(\left(\frac{7\sqrt{2}}{2}, \frac{7\sqrt{2}}{2}\right)\).

Step by step solution

01

Identify the given coordinates

The given polar coordinates are: \((r, \theta) = \left(7, \frac{\pi}{4}\right)\).
02

Apply the formula for x-coordinate

Use the formula for the x-coordinate: \(x = r \cos{\theta}\). Substitute the given values: \(x = 7 \cos{\left(\frac{\pi}{4}\right)}\).
03

Calculate the x-coordinate

Calculate \(x = 7 \cos{\left(\frac{\pi}{4}\right)}\). We know that \(\cos{\frac{\pi}{4}} = \frac{\sqrt{2}}{2}\). So, \(x = 7 \cdot \frac{\sqrt{2}}{2} = \frac{7\sqrt{2}}{2}\).
04

Apply the formula for y-coordinate

Use the formula for the y-coordinate: \(y = r \sin{\theta}\). Substitute the given values: \(y = 7 \sin{\left(\frac{\pi}{4}\right)}\).
05

Calculate the y-coordinate

Calculate \(y = 7 \sin{\left(\frac{\pi}{4}\right)}\). We know that \(\sin{\frac{\pi}{4}} = \frac{\sqrt{2}}{2}\). So, \(y = 7 \cdot \frac{\sqrt{2}}{2} = \frac{7\sqrt{2}}{2}\).
06

Write the rectangular coordinates

The rectangular coordinates are: \((x, y) = \left(\frac{7\sqrt{2}}{2}, \frac{7\sqrt{2}}{2}\right)\).

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