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Chapter 6: Problem 2

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(3,2^{1000} \pi\right)\)

Short Answer

Expert verified
The given polar coordinates \((3, 2^{1000}\pi)\) can be converted to rectangular coordinates using the conversion formulas. Since the angle \(\theta\) is an even multiple of \(\pi\), the angle lies on the positive x-axis, so \(\cos(\theta) = 1\) and \(\sin(\theta) = 0\). Applying the conversion formulas, we get the rectangular coordinates as \((x, y) = (3 \times 1, 3 \times 0) = (3, 0)\).

Step by step solution

01

Determine the angle in a simpler form

Since \(\theta = 2^{1000}\pi\), note that this is a multiple of \(\pi\). In fact, it is an even multiple of \(\pi\), which means the angle lies on the positive x-axis. Therefore, we know that: \(\cos(\theta) = 1\) \(\sin(\theta) = 0\) #Step 2: Apply the Conversion Formulas#
02

Calculate x and y using the conversion formulas

Using the given radius \(r = 3\) and the calculated values for \(\cos(\theta)\) and \(\sin(\theta)\), we can now apply the conversion formulas: \(x = r\cos(\theta) = 3 \times 1 = 3\) \(y = r\sin(\theta) = 3 \times 0 = 0\) #Step 3: Write the Rectangular Coordinates#
03

Write down the rectangular coordinates

The rectangular coordinates that correspond to the given polar coordinates \(\left(3,2^{1000} \pi\right)\) are: \((x, y) = (3, 0)\)

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