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Chapter 6: Problem 7

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(8, \frac{\pi}{3}\right)\)

Short Answer

Expert verified
The rectangular coordinates of the given point in polar coordinates \(\left(8, \frac{\pi}{3}\right)\) are \((4, 4\sqrt{3})\).

Step by step solution

01

Find x-coordinate

To find the x-coordinate, apply the formula \(x = r\cos{\theta}\) using the given polar coordinates: \[x = 8\cos\left(\frac{\pi}{3}\right)\] We know that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\), so the x-coordinate becomes: \[x = 8 \times \frac{1}{2} = 4\]
02

Find y-coordinate

To find the y-coordinate, apply the formula \(y = r\sin{\theta}\) using the given polar coordinates: \[y = 8\sin\left(\frac{\pi}{3}\right)\] We know that \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), so the y-coordinate becomes: \[y = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}\]
03

Combine x and y coordinates

Now that we have found the x and y coordinates, we can combine them to get the rectangular coordinates: \[(x, y) = (4, 4\sqrt{3})\] Thus, the rectangular coordinates of the given point in polar coordinates \(\left(8, \frac{\pi}{3}\right)\) are \((4, 4\sqrt{3})\).

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