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Chapter 6: Problem 3

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(4, \frac{\pi}{2}\right)\)

Short Answer

Expert verified
The rectangular coordinates of the given polar coordinates \(\left(4, \frac{\pi}{2}\right)\) are \((x, y) = (0, 4)\).

Step by step solution

01

Identify the polar coordinates

Let's identify the polar coordinates provided: \(r = 4\) \(\theta = \frac{\pi}{2}\)
02

Apply the conversion formula for the x-coordinate

Now, let's apply the conversion formula for the x-coordinate: \(x = r\cos(\theta) = 4\cos\left(\frac{\pi}{2}\right)\)
03

Evaluate the x-coordinate

We can evaluate the x-coordinate expression: \(x = 4\cos\left(\frac{\pi}{2}\right) = 4(0) = 0\)
04

Apply the conversion formula for the y-coordinate

Now, let's apply the conversion formula for the y-coordinate: \(y = r\sin(\theta) = 4\sin\left(\frac{\pi}{2}\right)\)
05

Evaluate the y-coordinate

We can evaluate the y-coordinate expression: \(y = 4\sin\left(\frac{\pi}{2}\right) = 4(1) = 4\)
06

Write the rectangular coordinates

With both the x and y values evaluated, we can now write the rectangular coordinates of the point: \((x, y) = (0, 4)\)

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Most popular questions from this chapter

Write \(2-2 i\) in polar form.

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Suppose \(z\) is a nonzero complex number. Show that \(\bar{z}=\frac{1}{z}\) if and only if \(|z|=1\).

Suppose \(w\) and \(z\) are complex numbers such that the real part of \(w z\) equals the real part of \(w\) times the real part of \(z\). Explain why either \(w\) or \(z\) must be a real number.
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