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Chapter 6: Problem 19

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(-6,0)$$

Short Answer

Expert verified
The polar coordinates of the point (-6,0) are (6, \(\pi\)).

Step by step solution

01

Find the value of 'r'

First, calculate the distance 'r' from the origin to the point using the formula \(r = \sqrt{x^{2} +y^{2}}\). With \(x = -6\) and \(y = 0\), this becomes \(r = \sqrt{(-6)^{2} + (0)^{2}} = \sqrt{36} = 6\). Therefore, 'r' is 6.
02

Find the value of '\(\theta\)'

Next, calculate the angle '\(\theta\)' Using the formula \(\theta = arctan\left (\frac{y}{x}\right )\). With \(x = -6\) and \(y = 0\), this becomes \(\theta = arctan\left (\frac{0}{-6}\right ) = arctan(0) = 0\). Therefore, '\(\theta\)' is 0. However, since x is negative, we are in the second or third quadrants. Since we are on the x-axis in this case, we need to change 0 to \(\pi\), because the angle is measured from the positive x axis.
03

Write the Point in Polar Coordinates

Now that values of 'r' and '\(\theta\)' have been obtained, they can be combined to write the point in polar coordinates. This results in the following: (6, \(\pi\))

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Most popular questions from this chapter

The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length of the major axis is \(2 a\) (see figure). Show that the polar equation of the orbit is \(r=a\left(1-e^{2}\right) /(1-e \cos \theta)\) where \(e\) is the eccentricity.

In Exercises 17-40, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=5\)

Sketch the graph of each equation. (a) \(r=3 \sec \theta\) (b) \(r=3 \sec \left(\theta-\frac{\pi}{4}\right)\) (c) \(r=3 \sec \left(\theta+\frac{\pi}{3}\right)\) (d) \(r=3 \sec \left(\theta-\frac{\pi}{2}\right)\)

In Exercises 29-32, use a graphing utility to graph the rotated conic. $$ r=\frac{2}{1-\cos (\theta-\pi / 4)} $$

In Exercises 83 and 84 , find the exact values of \(\sin 2 u\), \(\cos 2 u\), and \(\tan 2 u\) using the double-angle formulas. $$ \sin u=\frac{4}{5}, \frac{\pi}{2}
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