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Chapter 6: Problem 41

The rectangular coordinates of a point are given. Find polar coordinates of each point. Express \(\theta\) in radians. $$(-2,2)$$

Short Answer

Expert verified
The polar coordinates for the point (-2,2) are \((\sqrt{8}, 3蟺/4)\) radians.

Step by step solution

01

Calculate the distance r from the origin

To find r, we square each of the given coordinates, sum them, and then take the square root of the result using the formula \(r = \sqrt{x^2+y^2}\). Substituting x = -2 and y = 2 into this equation, we have \(r = \sqrt{(-2)^2+(2)^2} = \sqrt{8}\).
02

Calculate the angle 胃

To find 胃, we use the atan2 function in the formula \(胃 = atan2(y,x)\). Note that atan2 returns the result in radians. Substituting x = -2 and y = 2 into this equation, we get \( 胃 = atan2(2,-2)\). Now, apply the atan2 function. Atan2(2,-2) is equivalent to \(3蟺/4\), because the point (-2,2) lies in the second quadrant. The angle made with the positive part of the x-axis is \(3蟺/4\) radians.
03

Write the polar coordinates

The polar coordinates of a point are given as (r, 胃), where r is the distance from the origin to the point and 胃 is the angle from the positive x-axis. Our calculated values are \(r=\sqrt{8}\) and \(胃=3蟺/4\) radians. Thus, the polar coordinates for the point (-2,2) are \((\sqrt{8}, 3蟺/4)\).

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