Problem 34
Question
Convert the polar equation to rectangular form and sketch its graph. $$ r=2 \csc \theta $$
Step-by-Step Solution
Verified Answer
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.
1Step 1: 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 \(y = 2 \).
2Step 2: 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.
3Step 3: 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.
Other exercises in this chapter
Problem 34
Find the length of the curve over the given interval. $$ \begin{array}{ll} \text { Polar Equation } & \text { Interval } \\ \hline r=2 a \cos \theta & -\frac{\p
View solution Problem 34
Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. $$ x=\cos ^{2} \theta, \quad y=\cos \
View solution Problem 35
Find the length of the curve over the given interval. $$ \begin{array}{ll} \text { Polar Equation } & \text { Interval } \\ \hline r=1+\sin \theta a \cos \theta
View solution Problem 35
Determine the \(t\) intervals on which the curve is concave downward or concave upward. $$ x=t^{2}, \quad y=t^{3}-t $$
View solution