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Chapter 6: Problem 18

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(-3,-3)$$

Short Answer

Expert verified
The polar coordinates equivalent of the given rectangular coordinates (-3,-3) are \(\sqrt{18}, 5蟺/4\)

Step by step solution

01

Find the magnitude (r)

The magnitude or radial distance, r, is computed using the formula \(r = \sqrt{x^2 + y^2}\), where x and y are the given rectangular coordinates. Here, both x and y are -3, so \(r = \sqrt{(-3)^2 + (-3)^2} = \sqrt{18}\)
02

Find the angle (胃)

胃 is the angle in radians from the positive x-axis (counter-clockwise). This can be calculated using the formula \(胃 = arctan(y/x)\). If x < 0, add 蟺 to the result; if x > 0 and y < 0, add 2蟺 to the result. Since here both x and y are -3, \(胃 = arctan(-3/-3) = 蟺/4\). But since x < 0, we add 蟺 to the result, hence \(胃 = 5蟺/4\).
03

Write the final polar coordinates (r,胃)

The polar coordinates are usually written in the form (r,胃), so the final answer is \(\sqrt{18}, 5蟺/4\)

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Most popular questions from this chapter

Graphical Reasoning Use a graphing utility to graph the polar equation \(r=6[1+\cos (\theta-\phi)]\) for (a) \(\phi=0\), (b) \(\phi=\pi / 4\), and (c) \(\phi=\pi / 2\). Use the graphs to describe the effect of the angle \(\phi\). Write the equation as a function of \(\sin \theta\) for part (c).

\(r=-\frac{3 \pi}{4}\)

\(r=3 \sin 2 \theta\)

Consider the polar equation $$ r=\frac{4}{1-0.4 \cos \theta} $$ (a) Identify the conic without graphing the equation. (b) Without graphing the following polar equations, describe how each differs from the given polar equation. $$ r_{1}=\frac{4}{1+0.4 \cos \theta}, \quad r_{2}=\frac{4}{1-0.4 \sin \theta} $$ (c) Use a graphing utility to verify your results in part (b).

\(r^{2}=9 \cos 2 \theta\)
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