/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Convert the polar coordinates of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 32

Convert the polar coordinates of each point to rectangular coordinates. $$\left(\sqrt{3}, 150^{\circ}\right)$$

Short Answer

Expert verified
The rectangular coordinates are \(-\frac{3}{2}, \frac{ \sqrt{3} }{ 2 } \).

Step by step solution

01

- Understand Polar Coordinates

The polar coordinates are given as \(( r, \theta )\), where \( r \) represents the radius and \( \theta \) represents the angle in degrees from the positive x-axis.
02

- Convert Angle to Radians

Since the conversion formulas use radians, convert \(150^{\circ} \) to radians using the formula \( \theta_{\text{radians}}= \theta_{\text{degrees}} \times \frac{ \pi }{ 180 } \). Hence, \(150^{\circ} = 150 \times \frac{ \pi }{ 180 } = \frac{ 5 \pi }{ 6 } \) radians.
03

- Use Conversion Formulas

Use the formulas \[ x = r \cos( \theta ) \]\ and \[ y = r \sin( \theta ). \]\
04

- Substitute Values

Substitute \( r = \sqrt{3} \) and \( \theta = \frac{ 5 \pi }{ 6 } \) into the formulas: \[ x = \sqrt{3} \cos( \frac{ 5 \pi }{ 6 }) \], \[ y = \sqrt{3} \sin( \frac{ 5 \pi }{ 6 }). \]\
05

- Calculate x-coordinate

Calculate \( x \) using \(\frac{ 5 \pi }{ 6 } \): \[ x = \sqrt{ 3 } \cos( \frac{ 5 \pi }{ 6 } ). \] The cosine of \( \frac{ 5 \pi }{ 6 } \) is \(-\frac{ \sqrt{ 3 } }{ 2 } \), so \[ x = \sqrt{ 3 } \times \left( -\frac{ \sqrt{ 3 } }{ 2 } \right) = -\frac{3}{2}. \]
06

- Calculate y-coordinate

Calculate \( y \) using \(\frac{ 5 \pi }{ 6 } \): \[ y = \sqrt{ 3 } \sin( \frac{ 5 \pi }{ 6 } ). \] The sine of \( \frac{ 5 \pi }{ 6 } \) is \frac{ 1 }{ 2 }, so \[ y = \sqrt{ 3 } \times \frac{ 1 }{ 2 } = \frac{ \sqrt{3 } }{ 2 }. \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Polar Coordinates
Polar coordinates are a system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. In this system, the coordinates are written as \( (r, \theta) \).
  • \( r \) - This is the radius or the distance from the origin (which is the center point, usually considered as (0,0) in Cartesian coordinates).
  • \( \theta \) - This is the angle measured from the positive x-axis to the line connecting the origin to the point in question.
This method gives an alternative to the rectangular (or Cartesian) coordinates system which uses \( (x, y) \) pairs to mark positions on a plane.
Rectangular Coordinates
Rectangular coordinates (or Cartesian coordinates) describe a point in a plane using a pair of numbers (x, y). In this system, each point is defined by its horizontal distance from the y-axis and its vertical distance from the x-axis. Following this, we have:
  • x-coordinate 鈥� The horizontal distance from the y-axis.
  • y-coordinate 鈥� The vertical distance from the x-axis.
To convert polar coordinates \( (r, \theta) \) to rectangular coordinates \( (x, y) \), we use the formulas:
\( x = r \cos(\theta) \)
\( y = r \sin(\theta) \).
Angle Conversion
When working with trigonometric functions in mathematics, angles are most often required in radians rather than degrees. Radians provide a natural way of measuring angles based on the radius of a circle. To convert degrees to radians, we use the formula:
\( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \).
For example, converting \(150^{\circ}\) to radians: \(150^{\circ} \times \frac{ \pi }{ 180 } = \frac{ 5 \pi }{ 6 } \) radians.
This is crucial for using the trigonometric functions \( \cos(\theta) \) and \( \sin(\theta) \) correctly, as these functions require their input angle to be in radians.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write each vector as a linear combination of the unit vectors i and \(\mathbf{j}\). $$\langle 1,1\rangle$$

Find the angle to the nearest tenth of a degree between each given pair of vectors. $$\langle 2,1\rangle,\langle 3,5\rangle$$

For each given complex number, determine its complex conjugate in trigonometric form. $$3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$$

Find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\) for each pair of complex numbers, using trigonometric form. Write the answer in the form \(a+b i\). $$z_{1}=3 i, z_{2}=1+i$$

Determine whether each pair of vectors is parallel, perpendicular, or neither. $$\langle 2,3\rangle,\langle 8,12\rangle$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.