In Exercises 27-30 , rectangular coordinates of a point P are given. Use an algebraic method, an

In Exercises 27-30 , rectangular coordinates of a point P...

Key Concepts

Algebraic Methods and Approximation

Employing algebraic methods involves using equation-solving techniques to find the appropriate values for r and ? that satisfy the coordinate conversion equations. When exact analytical solutions are unwieldy, approximations using calculators become necessary to determine numerical values, particularly for trigonometric functions.

Angle Periodicity and Multiple Representations

Angles in polar coordinates are periodic, meaning that adding or subtracting full rotations (typically 2? radians) produces equivalent directions. This periodicity leads to multiple valid polar representations for a single point. Specifying different ranges for ?, such as [0, 2?] or [-?, ?], determines which of these representations are acceptable.

Rectangular Coordinate System

This is a coordinate system where each point in the plane is defined by an ordered pair (x, y), indicating horizontal and vertical displacements from an origin. Understanding these coordinates is essential for converting between different coordinate systems and interpreting positions in a two-dimensional plane.

Polar Coordinate System

In the polar coordinate system, points are represented by a radius r and an angle ?. The radius denotes the distance from the origin, while the angle measures the direction relative to the positive x?axis. This system is particularly useful for problems involving circular or rotational symmetry.

Coordinate Conversion Formulas

The conversion between rectangular and polar coordinates is achieved using the formulas r = ?(x² + y²) and tan(?) = y/x. These relationships provide a direct method for translating points from one coordinate system to another, and they enable the application of trigonometric principles to solve problems involving geometric positions.

Transcript

00:02 This is problem number 27, and we're given the coordinates of point p in rectangular coordinates, and we're asked to find its polar coordinates, r theta, and we're asked to do that in several different ways.

00:20 For parts a, b, and c, we're given different ranges of theta values, and so we'll list all possible answers within those ranges.

00:27 So the first thing i want to do is plot the point.

00:30 Get a feel for what we're looking at here.

00:32 We have the point 1 -1, and that would make a point.

00:36 A 45 degree angle with the horizontal.

00:41 So theta would be 45 degrees.

00:43 If we're answering in degrees, if we're answering in radiance, we would say pi over four radiance.

00:48 So we have that to start with, and we also need to find the value of r.

00:52 And we have the equation that relates the x and y coordinates to r, and that is x squared plus y squared equals r squared.

01:00 Substitute in one for x and one for y, and solve for r.

01:05 We get r squared equals two, and then.

01:07 So r equals the square root of 2.

01:10 So as we list the possible answers for our polar coordinates, we can use square root 2, or we can use negative square root 2 as our r value.

01:18 And if we use negative square root 2 as our r value, then we're going to need to go 180 degrees across to the other side, the other quadrant 3.

01:29 Okay, so one of our possible answers that falls between 0 and 2 pi is just the first answer we would naturally come up with, which is take the square root of two as your radius and 45 degrees as your angle.

01:43 Or if you're answering in radiance, the square root of two is your radius and pi over four as your angle.

01:49 And our other option, as i mentioned a minute ago, is to use negative square root two as your radius, and then you're going to need to add 180 degrees to that, and that would take you to 225 degrees.

02:00 So if we picture that on our graph over here, we have 225 degrees to take us all the way to quadrant 3.

02:09 If you want to convert that angle into radiance, you'll have r is negative square root 2, and you'll have 5 pi over 4 radians.

02:21 Now, for part b, we're looking for answers that fall between negative pi and pi radiance, so we can still use our original answer of square root 2, 45 degrees...

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