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Chapter 8: Problem 6

Use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. $$ (-2,11 \pi / 6) $$

Short Answer

Expert verified
The rectangular coordinates for the given point in polar coordinates (-2,11pi/6) are \(-sqrt(3), 1\).

Step by step solution

01

Understand the Problem

The point is given in polar coordinates (-2,11pi/6). Polar coordinates are represented as (r, theta) where r is the radius and theta is the angle. We have to convert these polar coordinates into rectangular coordinates (x, y). For this purpose we will use the formulas \(x = r * cos\(\theta)\) and \(y = r * sin\(\theta)\). Notice that \(r\) is negative.
02

Convert to Rectangular Coordinates

First, we replace \(r\) and \(\theta\) with their given values in the formulas for \(x\) and \(y\). This gives \(x = -2 * cos(11pi/6)\) and \(y = -2 * sin(11pi/6)\). Simplifying these expressions gives us the rectangular coordinates.
03

Calculate the Rectangular Coordinates

To find the exact values, recall that cos(11pi/6) is \(sqrt(3)/2\) and sin(11pi/6) is \(-1/2\). Substituting these values into the expressions gives \(x = -2* sqrt(3)/2 = -sqrt(3)\) and \(y = -2*(-1/2) = 1\). So the point in rectangular coordinates is \(-sqrt(3), 1\).

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Most popular questions from this chapter

In Exercises 49 and 50 , use the integration capabilities of a graphing utility to approximate to two decimal places the area of the region bounded by the graph of the polar equation. \(r=\frac{2}{3-2 \sin \theta}\)

Explain why finding points of intersection of polar graphs may require further analysis beyond solving two equations simultaneously

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Curtate cycloid: } x=2 \theta-\sin \theta, \quad y=2-\cos \theta $$

Show that the polar equation for \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta} \cdot \quad\) Ellipse

The curve represented by the equation \(r=a \theta,\) where \(a\) is a constant, is called the spiral of Archimedes. (a) Use a graphing utility to graph \(r=\theta,\) where \(\theta \geq 0\). What happens to the graph of \(r=a \theta\) as \(a\) increases? What happens if \(\theta \leq 0 ?\) (b) Determine the points on the spiral \(r=a \theta(a>0, \theta \geq 0)\) where the curve crosses the polar axis. (c) Find the length of \(r=\theta\) over the interval \(0 \leq \theta \leq 2 \pi\). (d) Find the area under the curve \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\).
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