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Chapter 6: Problem 25

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(6,9)$$

Short Answer

Expert verified
The point (6, 9) in rectangular coordinates converts to polar coordinates as \((\sqrt{117}, 0.9828)\).

Step by step solution

01

Calculate the radius

First, find the radius, which is the distance from the origin (0,0) to the point (6,9). Use the formula \(r = \sqrt{x^2 + y^2}\), plugging in \(x = 6\) and \(y = 9\) to get \(r = \sqrt{6^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117}\).
02

Calculate the angle

To find the angle, use the inverse tangent function, often known as arctan or \(\tan^{-1}\). The formula for the angle is \(\theta = \tan^{-1}(\frac{y}{x})\), so plug in \(y = 9\) and \(x = 6\) to get \(\theta = \tan^{-1}(\frac{9}{6})\). This is approximately 0.9828 radians.
03

Correct the quadrant of the angle

Since the point (6,9) is in the first quadrant, where \(x > 0\) and \(y > 0\), the angle doesn't need to be adjusted. Thus, the polar coordinates are given by the radius and the angle, written as \((r, \theta)\).

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Most popular questions from this chapter

Consider the equation \(r=3 \sin k \theta\). (a) Use a graphing utility to graph the equation for \(k=1.5\). Find the interval for \(\theta\) over which the graph is traced only once. (b) Use a graphing utility to graph the equation for \(k=2.5\). Find the interval for \(\theta\) over which the graph is traced only once. (c) Is it possible to find an interval for \(\theta\) over which the graph is traced only once for any rational number \(k\) ? Explain.

In Exercises 41-46, use a graphing utility to graph the polar equation. Describe your viewing window. \(r=8 \cos \theta\)

In Exercises 73-78, solve the trigonometric equation. $$ 9 \csc ^{2} x-10=2 $$

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

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Ellipse } & e=\frac{3}{4} & y=-3 \\ \end{array} $$
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