Problem 33 - Essential Calculus

Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=3 \sec \theta $$

Step-by-Step Solution

Answer

The rectangular form of \( r = 3 \ \sec \theta \) is \( x=3 \). The graph of this equation is a vertical line at \( x=3 \).

Step 1: Conversion from polar to rectangular form using identities

To convert this polar form to rectangular form, substitute \( r \cos \theta \) for x and \( r \sin \theta \) for y. But before that, implement the reciprocal identity for the secant function, \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, the equation becomes \( r = \frac{3}{\cos \theta} \). Now, simply substitute \( r \cos \theta \) for x in the equation, yielding \( r \cos \theta = 3 \). Implementing the fact that \( x = r \cos \theta \), the rectangular form equation is \( x=3 \).

Step 2: Sketching the rectangular form

The graph of \( x=3 \) is a vertical line that crosses the x-axis at \( x=3 \). This vertical line extends indefinitely in the positive and negative y direction.