Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Plot the point whose rectangular coondinates are given. Then determine nwo pairs of polar coondinates for the point with \(0^{\circ} \leq \theta<360^{\circ} .\) Do not use a calculator. $$\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right)$$

Step-by-Step Solution

Answer

The polar coordinates are \( (\sqrt{3}, 60^{\circ}) \) and \( (\sqrt{3}, 240^{\circ}) \).

Step 1: Understand Rectangular Coordinates

The given rectangular coordinates are \( \left( \frac{\sqrt{3}}{2}, \frac{3}{2} \right) \). This means the point is \( \frac{\sqrt{3}}{2} \) units along the x-axis and \( \frac{3}{2} \) units along the y-axis.

Step 2: Plot the Point on the Cartesian Plane

To plot \( \left( \frac{\sqrt{3}}{2}, \frac{3}{2} \right) \), locate \( \frac{\sqrt{3}}{2} \) on the x-axis and \( \frac{3}{2} \) on the y-axis. Place the point where these two values intersect.

Step 3: Calculate Radius \( r \) for Polar Coordinates

The formula to convert to polar coordinates is \( r = \sqrt{x^2 + y^2} \). Substitute the values:\[ r = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{9}{4}} = \sqrt{\frac{12}{4}} = \sqrt{3} \].

Step 4: Determine Angle \( \theta \) in Polar Coordinates

Use \( \tan \theta = \frac{y}{x} \) to find \( \theta \):\[ \tan \theta = \frac{\frac{3}{2}}{\frac{\sqrt{3}}{2}} = \frac{3}{\sqrt{3}} = \sqrt{3} \].The angle \( \theta \) whose tangent is \( \sqrt{3} \) is \( 60^{\circ} \). Thus, one pair of polar coordinates is \( (\sqrt{3}, 60^{\circ}) \).

Step 5: Find a Second Polar Coordinate Pair

Polar coordinates can be represented in multiple ways by adding \( 360^{\circ} \) to the angle. Hence, adding \( 360^{\circ} \) to \( 60^{\circ} \):\( (\sqrt{3}, 420^{\circ}) \).However, since \( 420^{\circ} \) is not within the desired range \( 0^{\circ} \leq \theta < 360^{\circ} \), subtract \( 360^{\circ} \):Thus, another pair is \( (\sqrt{3}, 240^{\circ}) \), using negative radius and supplementary angle \( 180^{\circ} \) from \( 60^{\circ} \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of representing points in a two-dimensional plane. They are denoted as \((x, y)\), where **x** is the horizontal distance from the origin, and **y** is the vertical distance.

• The coordinate pair \( \left( \frac{\sqrt{3}}{2}, \frac{3}{2} \right) \) tells you how far to move along the x-axis and y-axis, respectively.
• To understand rectangular coordinates, think of them as instructions. Go "this much" over and "this much" up.

Locating these points involves a simple system where each point's location is described by two numbers: one for horizontal movement and one for vertical movement. This system is widely used in mathematics and engineering because it provides a straightforward way to pin down exact locations of points in space.

Cartesian Plane

The Cartesian plane is a flat, two-dimensional surface defined by two intersecting lines: the x-axis (horizontal) and the y-axis (vertical). It is a crucial tool for plotting points and visualizing mathematical concepts.

• The origin, where the x-axis and y-axis intersect, is the point \((0, 0)\).
• Each axis divides the plane into four quadrants, labeled I, II, III, and IV.

In these quadrants, the signs of the x and y coordinates change:- In Quadrant I, both x and y are positive.- In Quadrant II, x is negative, y is positive.- In Quadrant III, both are negative.- In Quadrant IV, x is positive, y is negative.To plot the point \( \left( \frac{\sqrt{3}}{2}, \frac{3}{2} \right) \), you would locate \( \frac{\sqrt{3}}{2} \) along the x-axis and \( \frac{3}{2} \) along the y-axis. These coordinates place the point firmly in Quadrant I.

Coordinate Conversion

Coordinate conversion is the process of changing a point from one coordinate system to another. In this case, from rectangular coordinates to polar coordinates. This conversion allows you to express the same point as a distance from the origin and an angle of rotation.### From Rectangular to Polar CoordinatesTo convert rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\):1. Calculate the radius **r** using the formula: \[ r = \sqrt{x^2 + y^2} \] For our point, this becomes: \[ r = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \sqrt{3} \]2. Determine the angle **\(\theta\)** using tan: \[ \tan \theta = \frac{y}{x} \] Here, \( \tan \theta = \sqrt{3} \), giving us an angle \(\theta = 60^{\circ}\).These steps translate rectangular coordinates into polar ones, explaining how far and in what direction a point is from the origin. Such conversions are useful in various fields like physics and engineering, where rotational motion is often described in polar terms.