Example: Identify 4 Possible Polar Coordinates for a Point Using Radians
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
📍 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
💡Ordered Pair
💡Directed Distance
💡Directed Angle
💡Standard Position
💡Terminal Side
💡Co-Terminal Angles
💡Radians
💡Positive Angle
💡Negative Angle
💡Multiples of 2π Radians
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.
Transcripts
- IN THIS EXAMPLE WE WANT TO IDENTIFY EACH POINT
FOUR DIFFERENT WAYS USING POLAR COORDINATES.
IN POLAR COORDINATES A POINT IS EXPRESSED AS AN ORDERED PAIR,
WHERE THE FIRST COORDINATE IS R
AND THE SECOND COORDINATE IS THETA.
R IS THE DIRECTED DISTANCE FROM THE POLE,
AND THETA IS THE DIRECTED ANGLE FROM THE +X AXIS.
SO LET'S START BY LISTING TWO ORDERED PAIRS
FOR THE GIVEN POINT
WHEN R IS POSITIVE, OR R IS GREATER THAN ZERO.
SO IF R IS GOING TO BE POSITIVE
WE CAN JUST COUNT THE NUMBER OF UNITS FROM THE POLE
TO THE POINT, 1, 2, 3, 4, 5.
SO FOR BOTH OF THESE POINTS WE'LL LET R = 5.
AND IF R IS POSITIVE,
THAT MEANS THETA MUST BE AN ANGLE IN STANDARD POSITION
WHERE THE TERMINAL SIDE WOULD PASS THROUGH THIS POINT,
MEANING THE TERMINAL SIDE WOULD HAVE TO BE HERE,
IF R IS POSITIVE.
SO ANY ANGLE THAT HAS A TERMINAL SIDE HERE CAN BE USED FOR THETA.
SO IF WE WANT THETA TO BE THE LEAST POSITIVE ANGLE,
IT WOULD BE THE ANGLE FROM HERE TO HERE,
WHICH WE COULD SEE WOULD BE 180 + 45 DEGREES, OR 225 DEGREES.
IF WE WANT TO EXPRESS THIS IN RADIANS,
REMEMBER, PI/4 RADIANS = 45 DEGREES.
SO WE COULD COUNT, 1PI/4, 2PI/4, 3PI/4, 4PI/4, AND 5PI/4 RADIANS.
NOW, REMEMBER, THETA CAN BE ANY ANGLE
THAT'S CO-TERMINAL WITH THIS ANGLE,
SO IF WE WANTED ANOTHER POSITIVE CO-TERMINAL ANGLE,
WE COULD JUST ADD MULTIPLES OF 2 PI RADIANS TO THIS,
AND WE'D HAVE ANOTHER VALID ANGLE FOR THETA.
BUT IF WE WANT THETA TO BE NEGATIVE,
THE ANGLE WOULD HAVE TO ROTATE CLOCKWISE FROM HERE TO HERE,
WHICH WOULD BE -135 DEGREES, OR IN RADIANS,
AGAIN, COUNTING BY PI/4 RADIANS,
WE WOULD HAVE -1PI/4 RADIANS, -2PI/4 RADIANS,
AND THEN -3PI/4 RADIANS.
AND, OF COURSE, WE CAN SUBTRACT MULTIPLES OF 2PI RADIANS
FOR OTHER NEGATIVE ANGLES THAT WOULD BE CORRECT FOR THETA.
NOW LETS GO AHEAD AND LIST TWO ORDERED PAIRS
WHERE R IS NEGATIVE OR LESS THAN ZERO.
WELL, THIS POINT IS STILL 5 UNITS FROM THE POLE.
SO WE'LL LET R EQUAL -5 THIS TIME.
AND IF R IS NEGATIVE OR R IS LESS THAN ZERO,
THEN THE GIVEN POINT LIES ON THE RAY
POINTING IN THE OPPOSITE DIRECTION
OF THE TERMINAL SIDE OF THETA.
SO IF R IS NEGATIVE, THEN THE TERMINAL SIDE OF THETA
MUST POINT IN THE OPPOSITE DIRECTION,
MEANING IN THIS DIRECTION HERE.
SO ANY ANGLE THAT HAS A TERMINAL SIDE IN THIS LOCATION
WOULD BE CORRECT FOR THETA.
SO IF WE WANT THETA TO BE THE LEAST POSITIVE ANGLE,
IT WOULD ROTATE FROM HERE TO HERE, WHICH WOULD BE 45 DEGREES,
OR PI/4 RADIANS.
AND IF WE WANT THETA TO BE NEGATIVE,
WE'D HAVE A CLOCKWISE ROTATION FROM HERE TO HERE,
WHICH WOULD BE -315 DEGREES OR, AGAIN, COUNTING BY PI/4 RADIANS,
WE'D HAVE -1PI/4, -2PI/4, -3, - 4, -5, -6, AND -7PI/4 RADIANS.
AND, OF COURSE, ANY OTHER ANGLES THAT ARE CO-TERMINAL
TO THESE ANGLES WOULD ALSO BE CORRECT FOR THETA.
I HOPE YOU HAVE FOUND THIS HELPFUL.
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