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Chapter 10: Problem 30

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

Short Answer

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

Step by step solution

01

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 \(\frac{r}{\text{cos} \theta}= -4\).
02

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 \( x = -4 \).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 Ren茅 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 (胃).
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.

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Most popular questions from this chapter

A river has a constant current of \(3 \mathrm{~km} / \mathrm{h}\). At what angle to a boat dock should a motorboat capable of maintaining a constant speed of \(20 \mathrm{~km} / \mathrm{h}\) be headed in order to reach a point directly opposite the dock? If the river is \(\frac{1}{2}\) kilometer wide, how long will it take to cross?

Solve: \(\log _{4}(x+3)-\log _{4}(x-1)=2\).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(f(x)=x^{4},\) find \(\frac{f(x)-f(3)}{x-3}\)

A man pushes a wheelbarrow up an incline of \(20^{\circ}\) with a force of 100 pounds. Express the force vector \(\mathbf{F}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\).

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ 2 x^{2}+2 y^{2}=3 $$
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