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Chapter 6: Problem 15

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(-2.5,1.1)$$

Short Answer

Expert verified
The rectangular coordinates equivalent to the polar coordinates \((-2.5,1.1)\) are \((-1.12, 2.09)\)

Step by step solution

01

Calculate the x-coordinate

First, calculate the x-coordinate using the formula \(x = r \cdot \cos(\Theta)\). Here, \(r = -2.5\) and \(\Theta = 1.1\) radians. Plugging these values into the formula gives \(x = -2.5 \cdot \cos(1.1)\).
02

Compute the value of the x-coordinate

Calculating this gives approximately \(x = -1.12\).
03

Calculate the y-coordinate

Next, calculate the y-coordinate using the formula \(y = r \cdot \sin(\Theta)\). Here, \(r = -2.5\) and \(\Theta = 1.1\) radians. Plugging these values into the formula gives \(y = -2.5 \cdot \sin(1.1)\).
04

Compute the value of the y-coordinate

Calculating this yields approximately \(y = 2.09\).
05

Form the equivalent rectangular coordinates

The rectangular coordinates equivalent to the polar coordinates \((-2.5,1.1)\) are \((-1.12, 2.09)\).

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Most popular questions from this chapter

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Ellipse } & (20,0),(4, \pi) \\ \end{array} $$

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Hyperbola } & e=2 & x=1 \\ \end{array} $$

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In Exercises 73-78, solve the trigonometric equation. $$ \sqrt{2} \sec \theta=2 \csc \frac{\pi}{4} $$

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.
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