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Chapter 6: Problem 100

The rectangular coordinates of a point are given. Use a graphing utility in radian mode to find polar coordinates of each point to three decimal places. $$(-5,2)$$

Short Answer

Expert verified
The polar coordinates for the point (-5,2) to three decimal places are approximately (\(5.385, 3.524\)) in radian mode.

Step by step solution

01

Understand Rectangular and Polar Coordinate Systems

A point in the coordinate system can be represented in rectangular coordinates \((x, y)\) where 'x' is the horizontal distance and 'y' is the vertical distance from the origin. In the polar coordinates, the distance \(r\) from the origin and the angle \(\theta\) from the positive x-axis represents the same point. The relationship between the two systems can be expressed as \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}{\frac{y}{x}}\). The \(\tan^{-1}\) function must be applied carefully considering the quadrant where the point lies.
02

Calculate the radial coordinate \(r\)

Use the formula to calculate \(r\), \(r = \sqrt{x^2 + y^2}\). Substitute 'x' as '-5' and 'y' as '2' to get \(r = \sqrt{(-5)^2 + 2^2} = \sqrt{29} \approx 5.385\). This is the radial coordinate.
03

Calculate the angular coordinate \(\theta\)

The angular coordinate can be calculated as \(\theta = \tan^{-1}\frac{y}{x}\). Here it's crucial to consider the quadrant. '-5' lies in the negative x-axis so it is in the second quadrant. Thus, \(\theta = \tan^{-1}\frac{2}{-5} + \pi \approx \pi + 0.382 = \pi + 0.382 = 3.524\) radians. The \(\pi\) is added because the point is in the second quadrant.

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