/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Express the following Cartesian ... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 22

Express the following Cartesian coordinates in polar coordinates in at least two different ways. $$(-1,0)$$

Short Answer

Expert verified
Answer: The two polar coordinates representations for the given Cartesian coordinates (-1,0) are (1,蟺) and (1,3蟺).

Step by step solution

01

Calculate the radial distance

We will find the radial distance from the origin to the point using the formula: $$r = \sqrt{x^2 + y^2}$$ $$r = \sqrt{(-1)^2 + (0)^2}$$ $$r = \sqrt{1}$$ $$r = 1$$
02

Calculate the angle (first representation)

We'll calculate the angle \(\theta\) using the formula: $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ Since we have \(x=-1\) and \(y=0\), the angle is: $$\theta = \tan^{-1}\left(\frac{0}{-1}\right)$$ $$\theta = \tan^{-1}(0)$$ $$\theta = 0$$ Now we add $$\pi$$ to the angle to get the representation in the second quadrant: $$\theta = 0 + \pi$$ $$\theta = \pi$$ The first representation of the polar coordinates is: $$(r,\theta) = (1,\pi)$$
03

Calculate the angle (second representation)

Now we'll find another representation of the polar coordinates by adding $$2\pi$$ to the angle: $$\theta = \pi + 2\pi$$ $$\theta = 3\pi$$ The second representation of the polar coordinates is: $$(r,\theta) = (1,3\pi)$$
04

Final Answer

The two different polar coordinates representations for the given Cartesian coordinates \((-1,0)\) are: $$ (1, \pi) \quad \text{and} \quad (1, 3\pi) $$

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Most popular questions from this chapter

Sketch the graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$y^{2}=20 x$$

Give the equation in polar coordinates of a conic section with a focus at the origin, eccentricity \(e,\) and a directrix \(x=d,\) where \(d>0\)

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Find the area of the regions bounded by the following curves. The complete three-leaf rose \(r=2 \cos 3 \theta\)

Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{4}+\frac{y^{2}}{16}=1$$
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