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Chapter 8: Problem 31

In Exercises 21-40, convert each point given in polar coordinates to exact rectangular coordinates. $$ \left(2,240^{\circ}\right) $$

Short Answer

Expert verified
The point (2, 240掳) in polar coordinates converts to (-1, -鈭�3) in rectangular coordinates.

Step by step solution

01

Introduction to Polar and Rectangular Coordinates

Polar coordinates specify a point based on its distance from the origin and the angle from the positive x-axis. Rectangular coordinates (Cartesian) specify a point using the x-axis and y-axis.
02

Polar to Rectangular Conversion Formulas

To convert polar coordinates \(r, \theta\) to rectangular coordinates \(x, y\), we use the formulas: \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\). Here, \(r = 2\) and \(\theta = 240^\circ\).
03

Calculate x-coordinate

To find the x-coordinate, use the formula \(x = r \cdot \cos(\theta)\). Calculate \(\cos(240^\circ)\). Since \(240^\circ\) is in the third quadrant, it is equivalent to \(180^\circ + 60^\circ\), where \(\cos(60^\circ) = \frac{1}{2}\) and the cosine in the third quadrant is negative.\[\cos(240^\circ) = -\frac{1}{2}\]Substitute \(r = 2\) and \(\cos(240^\circ)\) into the formula:\[x = 2 \times -\frac{1}{2} = -1\]
04

Calculate y-coordinate

To find the y-coordinate, use the formula \(y = r \cdot \sin(\theta)\). Calculate \(\sin(240^\circ)\). Since \(240^\circ\) is in the third quadrant, it is equivalent to \(180^\circ + 60^\circ\), where \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\) and the sine in the third quadrant is negative.\[\sin(240^\circ) = -\frac{\sqrt{3}}{2}\]Substitute \(r = 2\) and \(\sin(240^\circ)\) into the formula:\[y = 2 \times -\frac{\sqrt{3}}{2} = -\sqrt{3}\]
05

Final Rectangular Coordinates

Combine the results from Step 2 and Step 3 to form the rectangular coordinates. The point \(2, 240^\circ\) in polar coordinates converts to \ (-1, -\sqrt{3}) in rectangular coordinates.

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Key Concepts

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

Understanding Cartesian Coordinates
The concept of Cartesian coordinates is fundamental in mathematics. It allows us to describe the position of points in a plane using two numbers associated with two perpendicular axes: the x-axis and the y-axis. Imagine a graph with a horizontal and a vertical line intersecting at the center. These lines are the axes, and their intersection point is known as the origin. Each point on this plane has a unique pair of values, referred to as coordinates, represented as \(x, y\).

  • The x-coordinate represents how far along the point is horizontally from the origin.
  • The y-coordinate indicates the vertical distance from the origin.
This system helps in pinpointing specific locations on a graph, making it easier to visualize functions or solve geometrical problems.
Mastering Coordinate Conversion
Coordinate conversion is all about switching between different systems for locating points. In the context of this exercise, we're converting polar coordinates to Cartesian coordinates. Polar coordinates use a distance from a central point (radius) and an angle from a reference direction.

To convert these into Cartesian coordinates, we rely on the formulas:
  • \(x = r \cdot \cos(\theta)\)
  • \(y = r \cdot \sin(\theta)\)
This method utilizes trigonometric functions to project a point from its polar description onto the Cartesian plane, effectively translating between the two different systems.
Demystifying Trigonometric Functions
Trigonometric functions, such as sine and cosine, are crucial tools in mathematics, especially in coordinate conversion. These functions help express relationships between the angles and sides of triangles. Cosine, \(\cos(\theta)\), relates to the horizontal aspect, while sine, \(\sin(\theta)\), relates to the vertical.

In our polar to rectangular conversion:
  • To determine the x-coordinate, we use the cosine function which projects the point's distance in line with the horizontal axis.
  • For the y-coordinate, the sine function evaluates the distance vertically from the horizontal axis.
This harmonious relationship between angle measurement and projection into coordinates allows seamless transformations and calculations in geometric space.

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Most popular questions from this chapter

In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=2 \sin (3 t), y=3 \cos (2 t), t \text { in }[0,2 \pi] $$

For Exercises 37-46, recall that the flight of a projectile can be modeled with the parametric equations $$ x=\left(v_{0} \cos \theta\right) t \quad y=-16 t^{2}+\left(v_{0} \sin \theta\right) t+h $$ where \(t\) is in seconds, \(v_{0}\) is the initial velocity in feet per second, \(\theta\) is the initial angle with the horizontal, and \(h\) is the initial height above ground, where \(x\) and \(y\) are in feet. Bullet Fired. A gun is fired from the ground at an angle of \(60^{\circ}\), and the bullet has an initial speed of 700 feet per second. How high does the bullet go? What is the horizontal (ground) distance between where the gun was fired and where the bullet hit the ground?

Given \(r=1+3 \cos \theta\), find the \(\theta\)-intervals for the inner loop.

For Exercises 47 and 48 , refer to the following: Modern amusement park rides are often designed to push the envelope in terms of speed, angle, and ultimately \(g\)-force, and usually take the form of gargantuan roller coasters or skyscraping towers. However, even just a couple of decades ago, such creations were depicted only in fantasy-type drawings, with their creators never truly believing their construction would become a reality. Nevertheless, thrill rides still capable of nauseating any would-be rider were still able to be constructed; one example is the Calypso. This ride is a not-too-distant cousin of the better-known Scrambler. It consists of four rotating arms (instead of three like the Scrambler), and on each of these arms, four cars (equally spaced around the circumference of a circular frame) are attached. Once in motion, the main piston to which the four arms are connected rotates clockwise, while each of the four arms themselves rotates counterclockwise. The combined motion appears as a blur to any onlooker from the crowd, but the motion of a single rider is much less chaotic. In fact, a single rider's path can be modeled by the following graph: The equation of this graph is defined parametrically by $$ \begin{aligned} &x(t)=A \cos t+B \cos (-3 t) \\ &y(t)=A \sin t+B \sin (-3 t), 0 \leq t \leq 2 \pi \end{aligned} $$ Amusement Rides. What is the location of the rider at \(t=0, t=\frac{\pi}{2}, t=\pi, t=\frac{3 \pi}{2}\), and \(t=2 \pi\) ?

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r^{2} \cos ^{2} \theta-2 r \cos \theta+r^{2} \sin ^{2} \theta=8 $$
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