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Chapter 6: Problem 25

Convert the polar coordinates of each point to rectangular coordinates. $$ (-3,3 \pi / 2) $$

Short Answer

Expert verified
The rectangular coordinates are (0, 3).

Step by step solution

01

Identify the polar coordinates

The given polar coordinates are \((-3, \frac{3\pi}{2})\). Here, \(r = -3\) and \(\theta = \frac{3\pi}{2}\).
02

Understand the relationship between polar and rectangular coordinates

To convert from polar to rectangular coordinates, use the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
03

Substitute the values into the formulas

Plug in \(r = -3\) and \(\theta = \frac{3\pi}{2}\) into the equations: \[x = -3 \cos\left(\frac{3\pi}{2}\right)\] and \[y = -3 \sin\left(\frac{3\pi}{2}\right)\].
04

Calculate the cosine and sine values

Recall that \(\cos\left(\frac{3\pi}{2}\right) = 0\) and \(\sin\left(\frac{3\pi}{2}\right) = -1\). Substituting these into the equations, we get: \[x = -3 \cdot 0 = 0\] and \[y = -3 \cdot (-1) = 3\].
05

Write the final rectangular coordinates

The rectangular coordinates are\((0, 3)\).

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Key Concepts

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

trigonometric functions
In order to convert from polar to rectangular coordinates, we need to use trigonometric functions. The two main functions here are cosine (\( \cos(\theta) \)) and sine (\( \sin(\theta) \)). These functions relate to the angles and sides of a right triangle.
When given an angle \( \theta \), \( \cos(\theta) \) gives the ratio of the adjacent side to the hypotenuse, and \( \sin(\theta) \) gives the ratio of the opposite side to the hypotenuse. In the unit circle:
  • \( \cos(0) = 1 \) and \( \sin(0) = 0 \)
  • \( \cos(\pi/2) = 0 \) and \( \sin(\pi/2) = 1 \)
  • \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \)
  • \( \cos(3\pi/2) = 0 \) and \( \sin(3\pi/2) = -1 \)
In our exercise, we use these functions to determine the x and y components of the rectangular coordinates.
coordinate systems
Understanding different coordinate systems is essential in math, physics, and engineering. The two main systems are:
  • Polar Coordinate System: Defined by a radius (r) and an angle (\( \theta \)).
  • Rectangular (Cartesian) Coordinate System: Defined by an x and y coordinate.
Polar coordinates are useful for problems involving circles or periodic functions. The coordinates are based on a distance from the origin and an angle from the positive x-axis. Rectangular coordinates, however, are more intuitive for most everyday applications, as they directly measure horizontal and vertical distances.
rectangular coordinates
Rectangular coordinates (also known as Cartesian coordinates) represent points in a plane using an x (horizontal) and y (vertical) coordinate. In our given exercise, we convert the polar coordinates \((-3, \frac{3\pi}{2})\) to rectangular coordinates. The formulas used for conversion are:
\( x = r \cos(\theta) \)
\( y = r \sin(\theta) \)
By substituting the values \( r = -3 \) and \( \theta = \frac{3\pi}{2} \), we find:
\( x = -3 \cos(\frac{3\pi}{2}) = 0 \)
\( y = -3 \sin(\frac{3\pi}{2}) = 3 \)
Hence, the rectangular coordinates are \((0, 3)\).
Understanding how to convert between these coordinate systems helps solve various mathematical problems more efficiently.

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Most popular questions from this chapter

Write each expression in the form \(a+\) bi where \(a\) and \(b\) are real numbers. $$ (4-5 i)^{2} $$

For each polar equation, write an equivalent rectangular equation. $$ r=\frac{1}{\sin \theta+\cos \theta} $$

For each rectangular equation, write an equivalent polar equation. $$ 2 x^{2}+y^{2}=1 $$

Convert the rectangular coordinates of each point to polar coordinates. Use radians for \(\theta .\) $$ (-3 \sqrt{2}, 3 \sqrt{2}) $$

Find the indicated roots. Express answers in the form \(a+b i\) The square roots of \(i\)
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