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Chapter 6: Problem 43

The rectangular coordinates of a point are given. Find polar coordinates of each point. Express \(\theta\) in radians. $$(2,-2 \sqrt{3})$$

Short Answer

Expert verified
The polar coordinates of the point are \((4, 5蟺 /3)\).

Step by step solution

01

Calculate the radial distance r

We first find the radial distance r from the origin to the point. This can be calculated as \(r = \sqrt{x^2+y^2}\), where x and y are the rectangular coordinates of the point. Plugging the given coordinates into this equation gives \(r = \sqrt{(2)^2+(-2\sqrt{3})^2}\). Simplifying this yields \(r = 4\).
02

Calculate the angle 胃

Next, we find the angle formed by the radial line and the positive x-axis using the equation \(胃 = atan2(y, x)\), where x and y are the rectangular coordinates of the point and atan2 is the two-argument arctangent function. Plugging the given coordinates into this equation gives \(胃 = atan2(-2\sqrt{3}, 2)\). This results in the angle \(胃 = - \pi / 3\). However, we note that in polar coordinates, r should be positive and 胃 should be represented in the range 0 鈮� 胃 < 2蟺. Adding 2蟺 to -蟺/3 results in \(胃 = 5蟺 /3\).
03

Write the polar coordinates

After finding the radial distance \(r = 4\) and the angle 胃 in the range 0 to 2蟺 as \(5蟺 /3\), write the polar coordinates out as \((4, 5蟺 /3)\)

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Most popular questions from this chapter

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=-6 \mathbf{i}$$

What are orthogonal vectors?

Will help you prepare for the material covered in the next section. Refer to Section 2.1 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i\). $$\frac{2+2 i}{1+i}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=2 \mathbf{i}+8 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j}$$

A force of 4 pounds acts in the direction of \(50^{\circ}\) to the horizontal. The force moves an object along a straight line from the point (3,7) to the point \((8,10),\) with distance measured in feet. Find the work done by the force.
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