Problem 30 - Algebra and Trigonometry

Question

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \sec \theta=-4 $$

Step-by-Step Solution

Answer

The equation transforms to \(x = -4\), a vertical line passing through \(x = -4\).

Step 1: Recall the definition of secant in terms of cosine

Recall that \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\). Therefore, \(r \text{sec} \theta = -4\) can be written as \(\frac{r}{\text{cos} \theta}= -4\).

Step 2: Convert to rectangular coordinates

We know that \(r \text{cos} \theta = x\). Hence, we substitute \(x\) in place of \(r \text{cos} \theta\) in the given equation. This converts \( \frac{r}{\text{cos} \theta} = -4 \) to \( x = -4 \).

Step 3: Identify and graph the equation

The equation \(x = -4\) represents a vertical line in the rectangular coordinate system that passes through \(x = -4 \). This line is parallel to the y-axis.

Key Concepts

rectangular coordinates

Rectangular coordinates are a way to represent points on the plane using two numbers: one for the horizontal position (the x-coordinate) and one for the vertical position (the y-coordinate). These coordinates are also referred to as Cartesian coordinates, named after the mathematician René Descartes.

polar coordinates

Polar coordinates represent points on the plane using a distance from a reference point (called the origin) and an angle from a reference direction (usually the positive x-axis). The distance is called the radial coordinate (r) and the angle is the angular coordinate (θ).

graphing equations

Graphing equations is about plotting points that satisfy a given mathematical equation on a coordinate system. These points can form lines, curves, or other shapes depending on the equation.