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Chapter 9: Problem 46

(a) \(r \cos \theta=-2\) (b) \(r \sin \theta=4\)

Short Answer

Expert verified
The point is (x, y) = (-2, 4) and \( r = 2\sqrt{5} \). \( \theta = \tan^{-1}(-2) \).

Step by step solution

01

Interpretation of Polar Coordinate Equations

In the problem, we are given equations involving polar coordinates: (a) \( r \cos \theta = -2 \) (b) \( r \sin \theta = 4 \).We need to convert these into Cartesian coordinates expressions, as well as find the value of \( r \) and \( \theta \).
02

Convert to Cartesian Coordinates

We know the relationships between polar and Cartesian coordinates: - \( x = r \cos \theta \)- \( y = r \sin \theta \).Use these to express the given equations in terms of \( x \) and \( y \):(a) \( x = -2 \)(b) \( y = 4 \).Thus, the Cartesian coordinate representation is simply \((x, y) = (-2, 4)\).
03

Find r and 胃

Using the equations, we know:1. \( r \cos \theta = -2 \)2. \( r \sin \theta = 4 \).Divide equation (2) by equation (1):\[ \frac{r \sin \theta}{r \cos \theta} = \frac{4}{-2} \]\[ \tan \theta = -2 \].Thus, \( \theta = \tan^{-1}(-2) \).To find \( r \), use the equation:\( r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2 \sqrt{5}\).

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Key Concepts

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

Cartesian Coordinates
Cartesian coordinates use two perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical). Each point in this coordinate system is represented as an ordered pair (x, y), where:
  • x is the horizontal position of the point.
  • y is the vertical position of the point.
These coordinates are particularly helpful because they allow for easy visualization and calculation of a point's location within a plane. In our daily lives, we often use Cartesian coordinates to map locations or describe positions in space.

In the context of the given exercise, we used the relationships:
  • For expression (a), the x-coordinate is derived: \(x = r \cos \theta = -2\).
  • For expression (b), the y-coordinate is calculated: \(y = r \sin \theta = 4\).
Thus, the Cartesian coordinates are represented as \((-2, 4)\). This straightforward approach is what makes Cartesian coordinates so useful in mathematical applications.
Conversion between Polar and Cartesian Coordinates
Polar coordinates represent a point in a plane using a distance and an angle. They are written as (r, \(\theta\)), where:
  • r is the distance from the origin to the point.
  • \(\theta\) is the angle measured from the positive x-axis to the line connecting the origin to the point.
To convert between polar and Cartesian coordinates, you utilize the following formulas:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
This is exactly what we did in the step-by-step solution. We transformed the given polar equations into Cartesian form. From these transformations:
  • Equation (a): \(x = -2\)
  • Equation (b): \(y = 4\)
This conversion is crucial in many fields such as physics, engineering, and computer graphics, where different kinds of calculations are required for various systems of coordinates.
Trigonometric Functions
Trigonometric functions relate the angles and sides of a right triangle. They are fundamental in various branches of mathematics and science. The primary trigonometric functions used in the exercise include:
  • Cosine (\(\cos\)): Relates the adjacent side to the hypotenuse in a right triangle.
  • Sine (\(\sin\)): Relates the opposite side to the hypotenuse.
  • Tangent (\(\tan\)): Ratio between sine and cosine, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
By understanding these functions, we can solve many kinds of problems including converting between coordinate systems.

In the exercise, we used these functions to find \(\theta\), specifically:
  • From \(\tan \theta = \frac{r \sin \theta}{r \cos \theta} = \frac{4}{-2} = -2\), we calculated \(\theta = \tan^{-1}(-2)\), which provides the angle relative to the x-axis.
This showcases how trigonometric functions serve as bridges between geometrical angles and algebraic coordinates.

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Most popular questions from this chapter

Graph the polar curve $$ r=1.5 \sin \left(\frac{30 \theta}{17}+\frac{\pi}{30}\right)+0.5 $$ Use a viewing rectangle extending from -2 to 2 in both the \(x\) - and \(y\) -directions. Let \(\theta\) run from 0 to \(34 \pi\)

Graph the polar curves. $$\begin{aligned} &\text { (a) } r=2|\sin \theta|^{\sin \theta}\\\ &\text { (b) } r=2|\sin \theta|^{\sin (\theta / 2)} \end{aligned}$$

Express vector in terms of the unit vectors i and \(\mathbf{j}.\) $$\langle 4,-2\rangle$$

You are given an angle \(\theta\) measured counterclockwise from the positive \(x\)-axis to a unit vector \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle\) In each case, determine the components \(u_{1}\) and \(u_{2}.\) $$\theta=3 \pi / 2$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=\cos t+t \sin t, y=\sin t-t \cos t(\text {involute of a circle})\) (a) \(-\pi \leq t \leq \pi\) (c) \(-4 \pi \leq t \leq 4 \pi\) (b) \(-2 \pi \leq t \leq 2 \pi\)
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