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Chapter 6: Problem 12

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(13, \frac{8 \pi}{3}\right)\)

Short Answer

Expert verified
The rectangular coordinates of the given polar coordinates \(\left(13, \frac{8\pi}{3}\right)\) are \(\left(-\frac{13}{2}, \frac{13\sqrt{3}}{2}\right)\).

Step by step solution

01

Find the cosine and sine of the angle

Find the cosine and sine of the angle \(\theta = \frac{8\pi}{3}\). Note that \(\frac{8\pi}{3}\) is equivalent to \(2\pi + \frac{2\pi}{3}\), which means it is co-terminal with the positive angle \(\frac{2\pi}{3}\). Therefore, we can find the cosine and sine of \(\frac{2\pi}{3}\) instead: \[\cos{\left(\frac{2\pi}{3}\right)} = -\frac{1}{2}\] \[\sin{\left(\frac{2\pi}{3}\right)} = \frac{\sqrt{3}}{2}\]
02

Multiply by the radius

Multiply the cosine and sine results from step 1 by the radius, \(r = 13\): \[x = 13 \cos{\left(\frac{2\pi}{3}\right)} = 13 \left(-\frac{1}{2}\right) = -\frac{13}{2}\] \[y = 13 \sin{\left(\frac{2\pi}{3}\right)} = 13 \left(\frac{\sqrt{3}}{2}\right) = \frac{13\sqrt{3}}{2}\]
03

Write the rectangular coordinates

Combine the \(x\) and \(y\) values calculated in step 2 to get the rectangular coordinates: \[(x, y) = \left(-\frac{13}{2}, \frac{13\sqrt{3}}{2}\right)\] So, the rectangular coordinates of the given polar coordinates \(\left(13, \frac{8\pi}{3}\right)\) are \(\left(-\frac{13}{2}, \frac{13\sqrt{3}}{2}\right)\).

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