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Chapter 11: Problem 20

Express the following polar coordinates in Cartesian coordinates. $$(4,5 \pi)$$

Short Answer

Expert verified
Question: Convert the polar coordinates (4, 5蟺) to Cartesian coordinates. Answer: The Cartesian coordinates are (4, 0).

Step by step solution

01

Identify the values of r and 胃

In the given polar coordinates \((4,5 \pi)\), we have: \(r = 4\) \(\theta = 5 \pi\)
02

Convert r and 胃 to x and y using conversion formulas

We will use the conversion formulas to find the Cartesian coordinates: \(x = r \cos(\theta)\) \(y = r \sin(\theta)\)
03

Calculate the x-coordinate

Using the conversion formula for x-coordinate and the given values of r and 胃, we get: \(x = 4 \cos(5\pi)\) Note that \(\cos(5\pi) = 1\) because cosine is positive for the even multiples (2, 4, 6...) of \(\pi\). So, \(x= 4 \cdot 1 =4\)
04

Calculate the y-coordinate

Using the conversion formula for y-coordinate and the given values of r and 胃, we get: \(y = 4 \sin(5 \pi)\) Note that \(\sin(5 \pi) = 0\) because sine is zero for all integer multiples of \(\pi\) (0, 1, 2...). So, \(y= 4 \cdot 0 =0\)
05

Final Answer

Therefore, the Cartesian coordinates of the given point \((4,5 \pi)\) in polar coordinates is \((4,0)\).

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Most popular questions from this chapter

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