Problem 5 - Calculus Early Transcendentals

Question

What is the polar equation of the vertical line \(x=5 ?\)

Step-by-Step Solution

Answer

Question: Convert the given Cartesian equation of a vertical line, \(x=5\), into a polar equation. Answer: The polar equation representing the same vertical line is \(r = \frac{5}{\cos(\theta)}\).

Step 1: Given Cartesian Equation

The given equation in Cartesian coordinates is \(x=5\).

Step 2: Convert to Polar Coordinates

To convert the Cartesian equation to a polar equation, we will use the conversion formulas: \(x = r \cos(\theta), y = r \sin(\theta)\). We want to find the polar equation that represents the same vertical line. So we can substitute \(x\) in the given equation with the corresponding polar coordinate expression: $$5 = r \cos(\theta)$$.

Step 3: Solve for 'r'

Now, we need to solve the equation for 'r': $$ r = \frac{5}{\cos(\theta)}$$.

Step 4: Polar Equation

The polar equation representing the same vertical line is: $$ r = \frac{5}{\cos(\theta)}$$.