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Chapter 6: Problem 44

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

Short Answer

Expert verified
The polar coordinates of the given point are \((4, \frac{5\pi}{6})\).

Step by step solution

01

Compute the radius r

The radius \(r\) is found using the Pythagorean theorem: \[ r=\sqrt{x^2+y^2} = \sqrt{(-2\sqrt{3})^2 + 2^2} = \sqrt{12+4} = \sqrt{16} = 4 \] Thus, the radius is \(r=4\).
02

Compute angle theta

The \(\theta\) is found by using the formula \[ \theta=\tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{2}{-2\sqrt{3}}\right) = -\frac{\pi}{6} \] However, since the point \(-2\sqrt{3},2\) is in the second quadrant, this value of \(\theta\) is not in the correct quadrant, so we add \(\pi\) to this to get the correct quadrant value of it. \[\theta = -\frac{\pi}{6} + \pi= \frac{5\pi}{6}\]
03

State the polar coordinates

After computing \(r\) and \(\theta\), the polar coordinates are found as \((r,\theta)=(4, \frac{5\pi}{6})\).

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Most popular questions from this chapter

Use a graphing utility to graph \(r=1+2 \sin n \theta\) for \(n=1,2,3,4,5,\) and \(6 .\) Use a separate viewing screen for each of the six graphs. What is the pattern for the number of large and small petals that occur corresponding to each value of \(n\) ? How are the large and small petals related when \(n\) is odd and when \(n\) is even?

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=\mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-3 \mathbf{j}$$

Solve the equation \(2 x^{3}+5 x^{2}-4 x-3=0\) given that -3 is a zero of \(f(x)=2 x^{3}+5 x^{2}-4 x-3\) (Section \(2.4,\) Example 6 )

Exercises \(81-83\) will help you prepare for the material covered in the next section. Two airplanes leave an airport at the same time on different runways. The first plane, flying on a bearing of \(\mathrm{N} 66^{\circ} \mathrm{W},\) travels 650 miles after two hours. The second plane, flying on a bearing of \(\mathrm{S} 26^{\circ} \mathrm{W},\) travels 600 miles after two hours. Illustrate the situation with an oblique triangle that shows how far apart the airplanes will be after two hours.

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+3 \mathbf{j}$$
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