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Chapter 6: Problem 5

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(6,-\frac{\pi}{4}\right)\)

Short Answer

Expert verified
The rectangular coordinates corresponding to the given polar coordinates \(\left(6, -\frac{\pi}{4}\right)\) are \((x, y) = (3\sqrt{2}, -3\sqrt{2})\).

Step by step solution

01

Write down the given polar coordinates

We are given polar coordinates \(\left(6, -\frac{\pi}{4}\right)\), where the radial distance \(r = 6\) and the polar angle \(\theta = -\frac{\pi}{4}\).
02

Convert the radial distance and polar angle to rectangular coordinates

Using the conversion equations, we find \(x\) and \(y\) as follows: \(x = r\cos\theta = 6\cos\left(-\frac{\pi}{4}\right)\) \(y = r\sin\theta = 6\sin\left(-\frac{\pi}{4}\right)\)
03

Evaluate the trigonometric functions

We now need to evaluate the cosine and sine functions for the angle \(-\frac{\pi}{4}\): \(\cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\) \(\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)
04

Calculate the rectangular coordinates

Now we can calculate the rectangular coordinates \(x\) and \(y\) by plugging the values of the cosine and sine functions: \(x = 6\left(\frac{\sqrt{2}}{2}\right) = 3\sqrt{2}\) \(y = 6\left(-\frac{\sqrt{2}}{2}\right) = -3\sqrt{2}\)
05

Write down the final answer

The rectangular coordinates corresponding to the given polar coordinates are \((x, y) = (3\sqrt{2}, -3\sqrt{2})\).

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