Lyons Little
08/05/2024 · Junior High School
The polar coordinates of a point are \( \left(8, \frac{3 \pi}{4}\right) \). Find the rectangular coordinates of this point.
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Step-by-step Solution
To convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use the following formulas:
\[
x = r \cos(\theta)
\]
\[
y = r \sin(\theta)
\]
Given the polar coordinates \((8, \frac{3\pi}{4})\), we have \(r = 8\) and \(\theta = \frac{3\pi}{4}\).
First, we calculate \(x\):
\[
x = 8 \cos\left(\frac{3\pi}{4}\right)
\]
The angle \(\frac{3\pi}{4}\) is in the second quadrant, where the cosine is negative. The cosine of \(\frac{3\pi}{4}\) is:
\[
\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]
Thus,
\[
x = 8 \left(-\frac{\sqrt{2}}{2}\right) = -4\sqrt{2}
\]
Next, we calculate \(y\):
\[
y = 8 \sin\left(\frac{3\pi}{4}\right)
\]
The sine of \(\frac{3\pi}{4}\) is positive in the second quadrant:
\[
\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
Thus,
\[
y = 8 \left(\frac{\sqrt{2}}{2}\right) = 4\sqrt{2}
\]
Now, we can summarize the rectangular coordinates:
\[
(x, y) = (-4\sqrt{2}, 4\sqrt{2})
\]
Therefore, the rectangular coordinates of the point are
\[
\boxed{(-4\sqrt{2}, 4\sqrt{2})}
\]
Quick Answer
The rectangular coordinates are \((-4\sqrt{2}, 4\sqrt{2})\).
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