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Chapter 8: Problem 34

Convert the polar equation to rectangular form and sketch its graph. $$ r=2 \csc \theta $$

Short Answer

Expert verified
The rectangular form of the equation \(r = 2csc(\theta)\) is \(y = 2\). The graph is a horizontal line passing through (0, 2) and extending indefinitely in both the positive and negative x-directions.

Step by step solution

01

Convert the Polar to Rectangular Form

\(\) First, we need to convert the polar equation to rectangular form. Given \(r = 2csc(\theta)\) and we know that \(csc(\theta) = 1/sin(\theta)\), our equation becomes \(r = 2/sin(\theta)\).Now, multiply each side by \(sin(\theta)\) to clear the fraction:\(r* sin(\theta) = 2 \)We know that \(y=r*sin(\theta)\), thus the rectangular equation is \(y = 2 \).
02

Graph the Equation

Next, we graph the equation \(y = 2\), which is a horizontal line passing through the point (0, 2) on the y-axis. It extends indefinitely in the positive and negative x-directions.
03

Analyze the Graph

After graphing the equation, we see that it corresponds to an infinite line along y=2 in the rectangular coordinate system. The polar coordinates (r, \(\theta\)) of any point on this line will satisfy the equation \(r = 2csc(\theta)\), showing that our conversion is correct.

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Most popular questions from this chapter

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{4}{1+2 \cos \theta}\)

Give the integral formulas for area and arc length in polar coordinates.

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. $$ \text { Witch of Agnesi: } x=2 \cot \theta, \quad y=2 \sin ^{2} \theta $$

Consider a projectile launched at a height \(h\) feet above the ground and at an angle \(\theta\) with the horizontal. If the initial velocity is \(v_{0}\) feet per second, the path of the projectile is modeled by the parametric equations \(x=\left(v_{0} \cos \theta\right) t\) and \(y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}\) The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an angle of \(\theta\) degrees with the horizontal at a speed of 100 miles per hour (see figure). (a) Write a set of parametric equations for the path of the ball. (b) Use a graphing utility to graph the path of the ball when \(\theta=15^{\circ} .\) Is the hit a home run? (c) Use a graphing utility to graph the path of the ball when \(\theta=23^{\circ} .\) Is the hit a home run? (d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run.

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=3 t-1, \quad y=2 t+1 $$
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