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Chapter 9: Problem 16

Find rectangular coordinates for the given polar point. $$\left(3, \frac{\pi}{8}\right)$$

Short Answer

Expert verified
The rectangular coordinates of the given polar point \((3, \frac{\pi}{8})\) are \((3 \cos(\frac{\pi}{8}), 3 \sin(\frac{\pi}{8}))\)

Step by step solution

01

Identify and extract the given values

The given values from the question are the radial distance \(r = 3\) and the angle \(\theta = \frac{\pi}{8}\)
02

Conversion of radial distance into x-coordinate

To find the x-coordinate, substitute the given values into the formula for the x-coordinate: \(x = r \cdot \cos(\theta)\), Therefore, \(x = 3 \cdot \cos(\frac{\pi}{8})\)
03

Conversion of radial direction into y-coordinate

Same goes for the y-coordinate, substitute the given values into the formula for the y-coordinate: \(y = r \cdot \sin(\theta)\). Hence, \(y = 3 \cdot \sin(\frac{\pi}{8})\)
04

Calculate the final rectangular coordinates

After performing the calculations in step 2 and 3, we get the rectangular coordinates: \(x\) and \(y\).

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Key Concepts

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

Polar to Cartesian Conversion
Converting polar coordinates to Cartesian coordinates is a practical mathematical skill. Let's break it down step by step to understand what is happening. In the polar coordinate system, a point is represented by \(r, \theta\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle in the polar plane. The aim is to transform these into the Cartesian form, which uses the familiar \(x, y\) coordinates.
  • The formula to convert the radial distance to an x-coordinate is \(x = r \cdot \cos(\theta)\).
  • Similarly, to convert the radial distance to a y-coordinate, use \(y = r \cdot \sin(\theta)\).
These transformations require the use of trigonometric functions, which we'll discuss next. The transformation helps in expressing the same point in a format often used in algebra and calculus.
Trigonometric Functions
Trigonometric functions are essential tools for converting between polar and Cartesian coordinates. These functions include sine \(\sin\) and cosine \(\cos\), among others, and they play a critical role in understanding angles and distances in mathematics.
  • Cosine \(\cos(\theta)\) measures the ratio of the adjacent side over the hypotenuse in a right-angled triangle.
  • Sine \(\sin(\theta)\) measures the ratio of the opposite side over the hypotenuse.
When converting polar coordinates, \(\cos(\theta)\) is used to find the horizontal (x) component, and \(\sin(\theta)\) is used to determine the vertical (y) component. Understanding these ratios is crucial since they allow for precise calculation of the Cartesian coordinates from polar information.
Coordinate System
A coordinate system is a framework that enables every point in a plane to be specified by a set of numbers. The Cartesian coordinate system is one of the most used systems in mathematics. It comprises two axes, \(x\) and \(y\), which intersect at a point called the origin.
  • The x-coordinate depicts the horizontal position.
  • The y-coordinate depicts the vertical position.
Conversely, the polar coordinate system specifies points with a radial distance and an angular direction.By understanding how these coordinate systems relate to each other, converting between them becomes more intuitive. This is particularly useful in fields like physics and engineering, where different types of data representation are needed.
Angle Measurement
Angle measurement is a fundamental concept essential for interpreting polar coordinates. In mathematics, angles can be measured in degrees or radians.
  • A full circle is 360 degrees, which is equivalent to \(2\pi\) radians.
  • The given angle \(\frac{\pi}{8}\) is a fraction of \(\pi\), indicative of radians, the SI unit for measuring angles.
Radians offer an efficient way of describing angles because they relate directly to the radius of a circle. Understanding this relation helps in using trigonometric functions effectively in polar to Cartesian conversion. In practice, converting angular measures between degrees and radians may be required for ease of calculation.

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Most popular questions from this chapter

In exercises parametric equations for the position of an object are given. Find the object's velocity and speed at the given times and describe its motion. $$ \left\\{\begin{array}{l} x=\cos 2 t \\ y=\sin \pi t \end{array}\right. $$

Find all points of intersection of the two curves. $$\left\\{\begin{array}{l}x=t+3 \\\y=t^{2}\end{array} \quad \text { and } \quad\left\\{\begin{array}{l}x=1+s \\\y=2-s\end{array}\right.\right.$$

Find parametric equations describing the given curve. The portion of the parabola \(y=x^{2}+1\) from (1,2) to (-1,2)

Explore the sound barrier problem discussed in the chapter introduction. Define 1 unit to be the distance traveled by sound in 1 second. Suppose that a jet has speed 0.8 unit per second (i.e., Mach 0.8 ) with position function \(x(t)=0.8 t\) and \(y(t)=0 .\) To model the position at time \(t=5\) seconds of various sound waves emitted by the jet, do the following on one set of axes. (a) Graph the position after 5 seconds of the sound wave emitted from (0,0) (b) graph the position after 4 seconds of the sound wave emitted from (0.8,0)\(;\) (c) graph the position after 3 seconds of the sound wave emitted from (1.6,0)\(;\) (d) graph the position after 2 seconds of the sound wave emitted from (2.4,0)\(;\) (e) graph the position after 1 second of the sound wave emitted from (3.2,0)\(;\) (f) mark the position of the jet at time \(t=5\)

In exercises parametric equations for the position of an object are given. Find the object's velocity and speed at the given times and describe its motion $$\left\\{\begin{array}{ll} x=40 t+5 & \text { (a) } t=0, \text { (b) } t=2 \\ y=20+3 t-16 t^{2} & \text { . } \end{array}\right.$$
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