Example: Identify 4 Possible Polar Coordinates for a Point Using Radians

Mathispower4u

7 Jun 201103:31

Summary

TLDRThis script explains how to represent a point in four different ways using polar coordinates. It covers scenarios where the radial distance (r) is positive or negative, and the angle (theta) is positive, negative, or zero. The explanation includes converting angles from degrees to radians and vice versa, and how to find co-terminal angles by adding multiples of 2π radians.

Takeaways

📍 Polar coordinates express a point as an ordered pair (r, θ), where r is the directed distance from the pole and θ is the directed angle from the +x-axis.
🔢 When r is positive, the point lies in the standard position where the terminal side of θ passes through the point.
🔄 The least positive angle for θ when r is positive can be calculated as 180° + 45° = 225°, or in radians, 5π/4.
🔁 Any co-terminal angle, which can be found by adding multiples of 2π radians to the least positive angle, is valid for θ when r is positive.
📉 If r is negative, the angle θ must rotate clockwise from the standard position to point towards the point.
⏲️ For a negative r, the least positive angle for θ is 45° or π/4 radians, and the negative angle would be -135° or -3π/4 radians.
🔀 Subtracting multiples of 2π radians from the negative angle gives other valid negative angles for θ.
🔄 When r is negative, the point lies on the ray pointing in the opposite direction of the terminal side of θ.
🔄 The least positive angle for θ when r is negative is 45° or π/4 radians, and the negative angle is -315° or multiples of -π/4 radians.
🔁 Any angle co-terminal with the calculated angles for θ is also correct when r is negative.

Q & A

What are polar coordinates?
-In polar coordinates, a point is expressed as an ordered pair (r, θ), where r represents the directed distance from the pole, and θ is the directed angle from the positive X-axis.
How do you determine the value of r when r is positive?
-When r is positive, you count the number of units from the pole to the point. In this example, r equals 5 because the point is 5 units away from the pole.
What is the value of θ when r is positive?
-When r is positive, θ is the angle in standard position where the terminal side passes through the point. The least positive angle for the point is 225 degrees or 5π/4 radians.
How can you find other possible values for θ when r is positive?
-You can add multiples of 2π radians to the base angle to find co-terminal angles. For example, adding 2π to 5π/4 radians will give you another valid angle for θ.
What is the negative value of θ when r is positive?
-The negative value of θ would be a clockwise rotation from the positive X-axis to the point. This gives a value of -135 degrees or -3π/4 radians.
How do you find negative angles co-terminal with a given negative θ?
-To find co-terminal negative angles, subtract multiples of 2π radians from the given negative angle.
What happens when r is negative?
-When r is negative, the point lies on the ray pointing in the opposite direction of the terminal side of θ. For example, r = -5 means the point is still 5 units away from the pole but in the opposite direction.
What is the value of θ when r is negative?
-When r is negative, the least positive angle for θ would be 45 degrees or π/4 radians, as the terminal side must point in the opposite direction.
What is the negative value of θ when r is negative?
-The negative value of θ when r is negative would be a clockwise rotation to the point, which gives -315 degrees or -7π/4 radians.
Can you have multiple values for θ when r is negative?
-Yes, any angle co-terminal with the base angle is valid. You can find these angles by adding or subtracting multiples of 2π radians.

Outlines

00:00

📍 Polar Coordinates Explanation

The paragraph introduces polar coordinates as an alternative method to represent points in a two-dimensional plane. It explains that a point in polar coordinates is an ordered pair with the first element, r, representing the directed distance from the pole (origin) and the second element, theta, representing the directed angle from the positive x-axis. The paragraph then proceeds to illustrate how to determine these values for a specific point. It explains that if r is positive, the point is located in the standard position with the terminal side passing through the point. The least positive angle for theta is calculated as 225 degrees or 5π/4 radians, and it's noted that any co-terminal angle (angles that differ by full rotations of 2π radians) can also be used for theta. Conversely, if r is negative, the point is on the opposite ray of theta's terminal side. The least positive angle for theta in this case is 45 degrees or π/4 radians, and negative angles are obtained by rotating clockwise from the positive x-axis.

Mindmap

Keywords

💡Polar Coordinates

Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point, called the pole, and an angle from a reference direction, typically the positive x-axis. In the video, polar coordinates are used to identify points in different ways. The script explains that a point is expressed as an ordered pair (r, θ), where r is the directed distance from the pole and θ is the directed angle from the +x axis.

💡Ordered Pair

An ordered pair is a pair of numbers that are usually written within parentheses and are ordered in a sequence. In the context of polar coordinates, the ordered pair (r, θ) represents a point in the plane, where 'r' is the radius or distance from the origin (pole) and 'θ' is the angle (theta). The script uses ordered pairs to list different ways to express a point's location using polar coordinates.

💡Directed Distance

Directed distance, or 'r' in polar coordinates, refers to the distance of a point from the pole (origin) and includes direction. It is positive if the point is in the counterclockwise direction from the +x axis and negative if it is in the clockwise direction. The script explains that when r is positive, the point is a certain number of units from the pole, and when r is negative, the point lies in the opposite direction.

💡Directed Angle

Directed angle, or 'θ' (theta), is the angle measured from the positive x-axis in a counterclockwise direction. It indicates the direction from the pole to the point. The script discusses how θ can vary depending on the position of the point and whether r is positive or negative, including examples of angles in standard position and co-terminal angles.

💡Standard Position

Standard position refers to the initial orientation of the angle where the initial side is on the positive x-axis. In the script, standard position is mentioned when discussing how the terminal side of an angle would pass through a point if r is positive, meaning the angle is measured from the positive x-axis.

💡Terminal Side

The terminal side of an angle is the line that forms the angle with the initial side. In polar coordinates, the terminal side of the angle θ determines the direction from the pole to the point. The script uses the concept of the terminal side to explain how the angle changes depending on the direction of r (positive or negative).

💡Co-Terminal Angles

Co-terminal angles are angles that share the same terminal side. They differ by full rotations (multiples of 360 degrees or 2π radians). The script explains that θ can be any co-terminal angle, meaning you can add or subtract multiples of 2π radians to get another valid angle for θ.

💡Radians

Radians are a unit of angular measure where the angle of one radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. The video script converts degrees to radians to express angles in a mathematically precise way, using the conversion that π/4 radians equals 45 degrees.

💡Positive Angle

A positive angle is measured in a counterclockwise direction from the positive x-axis. In the context of the script, a positive angle for θ is used when r is positive, indicating the direction from the pole to the point in a counterclockwise manner.

💡Negative Angle

A negative angle is measured in a clockwise direction from the positive x-axis. The script mentions negative angles for θ when r is negative, showing how the angle would rotate clockwise from the positive x-axis to reach the point.

💡Multiples of 2π Radians

Adding or subtracting multiples of 2π radians to an angle results in co-terminal angles. The script uses this concept to explain how you can find different valid angles for θ by adding or subtracting these multiples, whether the angles are positive or negative.

Highlights

Introduction to identifying points using polar coordinates.

Polar coordinates expressed as an ordered pair with r and theta.

r represents the directed distance from the pole.

Theta represents the directed angle from the +x axis.

Listing ordered pairs for points with positive r values.

Example of r being 5 units from the pole.

Positive r implies theta is in standard position.

Least positive angle for theta is 225 degrees or 5π/4 radians.

Co-terminal angles can be found by adding multiples of 2π radians.

Negative r values and their implications for theta.

Example of r being -5 with the point still 5 units from the pole.

Negative r means the point lies on the opposite direction of theta's terminal side.

Least positive angle for theta with negative r is 45 degrees or π/4 radians.

Negative theta can be found by rotating clockwise from the positive x-axis.

Co-terminal angles for negative theta can be found by subtracting multiples of 2π radians.

Summary of how to find polar coordinates for points with both positive and negative r values.

The importance of understanding co-terminal angles in polar coordinates.

Practical applications of polar coordinates in various fields.

The transcript's conclusion and the hope for its helpfulness.