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Chapter 6: Problem 35

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$\left(2, \frac{\pi}{3}\right)$$

Short Answer

Expert verified
The rectangular coordinates of the point are (1, \(\sqrt{3}\))

Step by step solution

01

Identify the polar coordinates

Polar coordinates are given in the format \((r, \theta)\). In this case, they are \((2, \frac{\pi}{3})\) where \(r = 2\) is the distance from the origin and \(\theta = \frac{\pi}{3}\) is the angle in radian.
02

Calculate the x-coordinate

The x-coordinate in rectangular coordinates can be calculated from polar coordinates using the formula: \(x = r \cdot \cos(\theta)\). Substituting the given values into the formula to find the x-coordinate: \(x = 2 \cdot \cos(\frac{\pi}{3})\). This will give the x-coordinate value as 1.
03

Calculate the y-coordinate

The y-coordinate in rectangular coordinates can be calculated from polar coordinates using the formula: \(y = r \cdot \sin(\theta)\). Substituting the given values into the formula to find the y-coordinate: \(y = 2 \cdot \sin(\frac{\pi}{3})\). This will give the y-coordinate value as \(\sqrt{3}\).

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