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Polar Coordinates

Study Guide

Key Definition

Polar coordinates $(r, \theta)$ describe the point at distance $r$ from the origin at an angle $\theta$ from the x-axis.

Important Notes

  • Distance $r$ can be positive or negative; a negative $r$ corresponds to the point at |r| with angle $\theta + 180^\\circ$ (or $\pi$ radians).
  • Angle $\theta$ can be measured in degrees or radians; keep units consistent.
  • To convert to Cartesian: $x = r \cos(\theta)$, $y = r \sin(\theta)$.
  • To convert to Polar: $r = \sqrt{x^2 + y^2}$, $\theta = atan2(y, x)$ (accounts for the correct quadrant).
  • Use the correct angle unit

Mathematical Notation

$\theta$ represents the angle in polar coordinates$r$ represents the radial distance$\sin$ is the sine function$\cos$ is the cosine function$\tan$ is the tangent functionRemember to use proper notation when solving problems

Why It Works

Polar coordinates provide a natural way to express positions in circular and rotational contexts using $r$ and $\theta$.

Remember

The conversion formulas connect polar and Cartesian coordinates using trigonometric relationships.

Quick Reference

Cartesian to Polar:$r = \sqrt{x^2 + y^2}$, $\theta = \tan^{-1}(\frac{y}{x})$Polar to Cartesian:$x = r \cos(\theta)$, $y = r \sin(\theta)$

Understanding Polar Coordinates

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Beginner Explanation

In polar coordinates, a point is described by $(r, \theta)$ where $r$ tells you how far to go from the origin and $\theta$ tells you which direction to go measured from the positive x-axis.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Convert the polar coordinates $(5, 45^\\circ)$ to Cartesian coordinates.

A$(3.54, 3.54)$B$(5.00, 0.00)$C$(0.00, 5.00)$D$(2.50, 4.33)$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

Imagine you are plotting a point on a radar screen at $r = 10$ and $\theta = 30^\\circ$. Convert this to Cartesian coordinates to find the exact position.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Given a point $(-3, 3)$ in Cartesian coordinates, find its polar coordinates.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Convert the Cartesian coordinates $(-4, 4\sqrt{3})$ to polar coordinates.

A$(8, 120^\\circ)$B$(8, 60^\\circ)$C$(4, 90^\\circ)$D$(8, 150^\\circ)$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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