Problem 5 - A Graphical Approach to Precalculus with Limits

Question

For each point given in polar coordinates, state the quadrant in which the point lies if it is graphed in a rectangular coordinate system. (a) \(\left(5,135^{\circ}\right)\) (b) \(\left(2,60^{\circ}\right)\) (c) \(\left(6,-30^{\circ}\right)\) (d) \(\left(4.6,213^{\circ}\right)\)

Step-by-Step Solution

Answer

(a) Quadrant II, (b) Quadrant I, (c) Quadrant IV, (d) Quadrant III.

Step 1: Convert Polar Coordinates to Rectangular Angles

Polar coordinates are given in angles, and these angles can be directly used to identify quadrants in the rectangular coordinate system. A full circle is 360 degrees, so each quadrant is 90 degrees. Quadrant I is from 0 to 90 degrees, Quadrant II is from 90 to 180 degrees, Quadrant III is from 180 to 270 degrees, and Quadrant IV is from 270 to 360 degrees.

Step 2: Identify Quadrants for (5, 135^{\circ})

Since 135° is between 90° and 180°, the point \(5, 135^{\circ}\) lies in Quadrant II.

Step 3: Identify Quadrants for (2, 60^{\circ})

60° is within the 0° to 90° range, so the point \(2, 60^{\circ}\) is in Quadrant I.

Step 4: Identify Quadrants for (6, -30^{\circ})

An angle of -30° is equivalent to adding 360°, which results in 330°. Since 330° lies between 270° and 360°, the point \(6, -30^{\circ}\) is in Quadrant IV.

Step 5: Identify Quadrants for (4.6, 213^{\circ})

213° is between 180° and 270°, so the point \(4.6, 213^{\circ}\) falls in Quadrant III.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a foundational concept in geometry used to locate points in a plane using an ordered pair of numbers.

• The system is defined by two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical).
• These axes intersect at a point called the origin, denoted as \( (0, 0) \).

Each point in this system is expressed as \( (x, y) \), where \( x \) is the horizontal distance from the origin, and \( y \) is the vertical distance.The plane is divided into four sections, known as quadrants, which help to describe the location of points more specifically.

Coordinates Conversion

To work with different types of coordinate systems, such as polar and rectangular, it's essential to understand how to convert between them.- **Polar Coordinates**: These are given as \( (r, \theta) \), where \( r \) is the radius (distance from the origin), and \( \theta \) is the angle from the positive x-axis.- **Rectangular Coordinates**: As mentioned earlier, this system uses \( (x, y) \).The conversion from polar to rectangular coordinates requires some basic trigonometry:

• To find \( x \): Use the formula \( x = r \cos(\theta) \)
• To find \( y \): Use the formula \( y = r \sin(\theta) \)

These conversions are crucial when it comes to graphing polar coordinates on a rectangular graph or vice versa.

Quadrants Identification

Identifying the correct quadrant for a point is vital in understanding its position in the rectangular coordinate system.

• **Quadrant I**: This ranges from 0° to 90° and both \( x \) and \( y \) coordinates are positive.
• **Quadrant II**: From 90° to 180°, where \( x \) is negative and \( y \) is positive.
• **Quadrant III**: Between 180° and 270°, \( x \) and \( y \) are both negative.
• **Quadrant IV**: From 270° to 360°, where \( x \) is positive and \( y \) is negative.

This system helps accurately determine the graphing position of a point when given an angle in the polar system.

Angle Measurement

Angle measurement is crucial in both polar and rectangular coordinate systems. It determines the direction of a point from the origin.- **Degrees**: The most common unit of measurement. A full circle is divided into 360 degrees.- **Radian**: Another unit often used in mathematics, particularly in calculus and trigonometry, where one full circle equals \( 2\pi \) radians.When dealing with problems that involve negative angles, such as -30°, it helps to convert this to a positive equivalent by adding 360° to ensure it fits within the standard range, making calculations consistent.Understanding these measurements is key when determining the corresponding quadrant or when converting between coordinate systems.