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Chapter 6: Problem 10

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(11,-\frac{\pi}{6}\right)\)

Short Answer

Expert verified
The rectangular coordinates for the given polar coordinates \(\left(11,-\frac{\pi}{6}\right)\) are: \[ (x, y) = \left(11 \cdot \frac{\sqrt{3}}{2}, 11 \cdot -\frac{1}{2}\right) \]

Step by step solution

01

Calculate the x-coordinate

To find the x-coordinate, we will use the formula \(x = r\cos(\theta)\). In our case, \(r = 11\) and \(\theta = -\frac{\pi}{6}\). Thus, \[x = 11 \cos\left(-\frac{\pi}{6}\right)\]
02

Calculate the x-coordinate value

We know that \(\cos(-\theta) = \cos(\theta)\) and \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\), so \[x = 11 \cdot \frac{\sqrt{3}}{2}\]
03

Calculate the y-coordinate

To find the y-coordinate, we use the formula \(y = r\sin(\theta)\). With \(r = 11\) and \(\theta = -\frac{\pi}{6}\), we get \[y = 11\sin\left(-\frac{\pi}{6}\right)\]
04

Calculate the y-coordinate value

We know that \(\sin(-\theta) = -\sin(\theta)\) and \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\). So, \[y = 11 \cdot -\frac{1}{2}\]
05

Write the final rectangular coordinates

Putting the x-coordinate and y-coordinate together, we get the rectangular coordinates as \[ (x, y) = \left(11 \cdot \frac{\sqrt{3}}{2}, 11 \cdot -\frac{1}{2}\right) \]

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