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Chapter 6: Problem 42

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 of the given point are \((2\sqrt{2}, -\frac{蟺}{4})\).

Step by step solution

01

Calculation of r

Firstly, we need to calculate 'r' which is the distance from the origin (0,0) to the given point (2,-2). We can calculate 'r' by using the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \), where 'x' is the x-coordinate and 'y' is the y-coordinate. Therefore, \( r = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2} \).
02

Calculation of 胃

Secondly, we need to find the angle '胃'. This can be done by using the inverse tangent function: \( 胃 = \arctan(\frac{y}{x}) \), where 'y' is the y-coordinate and 'x' is the x-coordinate. Please note that the sign of 'x' and 'y' are very important in deciding the quadrant. Here, 'x' is positive and 'y' is negative. This places the point in the fourth quadrant where 胃 is negative. Hence, \( 胃 = \arctan(\frac{-2}{2}) = -\frac{蟺}{4} \). To express 胃 in radians, please make sure the negative sign is kept. Since 胃 is already in radians, there is no need for conversion.

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