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Chapter 6: Problem 45

The rectangular coordinates of a point are given. Find polar coordinates of each point. Express \(\theta\) in radians. $$(-\sqrt{3},-1)$$

Short Answer

Expert verified
The polar coordinates of the point with rectangular coordinates (-\sqrt{3}, -1) are given by the radial coordinate r and the angle \(\theta\) in radians.

Step by step solution

01

Identify the rectangular coordinates

Given that the rectangular coordinates are (-\sqrt{3}, -1). These are represented as (x, y).
02

Calculate the radial coordinate

The radial coordinate r in polar coordinates can be calculated by the formula r = \sqrt{x^2 + y^2}, which gives \( r = \sqrt{(-\sqrt{3})^2 + (-1)^2} \).
03

Determine the angle \(\theta\)

The angle \(\theta\) is given by the formula \(\theta = \arctan(\frac{y}{x})\). Substituting for \(x = -\sqrt{3}\) and \(y = -1\), we calculate for the value of \(\theta\).
04

Adjust the angle for the correct quadrant

Since the point (-\sqrt{3}, -1) lies in the third quadrant, we have to add \(\pi\) to the calculated \(\theta\) since \(\arctan\) only gives values in the first and fourth quadrants.
05

Confirm the results

Verify that the polar coordinates found represent the same point in the plane as the original rectangular coordinates.

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Most popular questions from this chapter

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+\frac{18}{5} \mathbf{j}$$

Use the vectors $$\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad \text { and } \quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j},$$ to prove the given property. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=3 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}+5 \mathbf{j}$$

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 \(a+b i\). $$\frac{2+2 i}{1+i}$$

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 \(a+b i\). $$(-1+i \sqrt{3})(-1+i \sqrt{3})(-1+i \sqrt{3})$$
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