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

Rectangular to Polar Coordinate Conversion

This concept involves the relationship between rectangular (Cartesian) coordinates (x, y) and polar coordinates (r, ?). The conversion is done using the formulas r = ?(x² + y²) and ? = arctan(y/x), with additional adjustments made based on the sign of x and y in order to determine the correct angle. This fundamental method is used to transform any point in the rectangular plane to its equivalent polar representation.

Multiple Representations Due to Periodicity

Polar coordinates are not unique because the angle ? can be expressed in infinitely many ways by adding or subtracting full rotations (multiples of 2?). For examining solutions within different specified ranges for ?, the periodic nature of trigonometric functions is exploited, which means the same point may have multiple polar coordinate representations depending on the domain constraints imposed on ?.

Quadrant Consideration and Angle Adjustment

When converting from rectangular to polar coordinates, it's critical to identify the quadrant in which the point lies since the inverse tangent function typically returns values in a restricted interval. This requires adjusting the computed angle to ensure that it correctly represents the direction of the point, taking into account the signs of x and y. This adjustment guarantees that the resulting angle appropriately corresponds to the location of the point in the coordinate system.

Transcript

00:01 Okay, this is problem number 29, and we're given point p with coordinates negative 25, and those are rectangular coordinates, and we're asked to find the polar coordinates of the point in as many ways as possible, depending on part a, b, or c, what kind of range they give us for theta.

00:18 So the first thing i'd like to do is plot the point.

00:20 So we have negative 25, so that would be roughly right here.

00:27 And so we want to figure out the angle that that forms.

00:33 So in order to do that, we can drop a perpendicular from the point.

00:36 Create a reference triangle and the leg here is 2 and that leg there is 5.

00:42 And so in order to find the measure of the angle we can use the tangent ratio, tangent of theta.

00:49 Now in quadrant 2 tangent is negative, so keep that in mind.

00:53 So tangent of theta equals negative 5 over 2, negative opposite over adjacent.

00:58 Now this is not one that you would naturally know the exact value for.

01:01 You're going to have to use your calculator on this one and approximate.

01:06 So first we find the inverse tangent of negative five halves.

01:10 Use your calculator for that.

01:12 Now, i would assume that we should answer in radiance for this problem because of the way the ranges were given between 0 and 2 pi.

01:19 However, i noticed that the answer in the book is given in degrees.

01:22 So what i'm going to do just to be safe is give the degree answers, as well as the radiant answers, and you can choose whichever column of answers you'd like.

01:30 Okay, so using my calculator, the inverse tangent of negative 5 halves, is approximately negative 1 .19 radiance.

01:38 Now, the calculator will give you the answer in that format, but in order to get something that's between 0 and 2 pi, we need to go ahead and add pi to that, and so that answer is going to be 1 .95 radiance.

01:53 Now, if you convert that to degrees, or if you just go through the calculator process again and ask your calculator to find the degree measure, it's going to give you negative 68 .2 degrees, but we're going to add 180 degrees to that, to get it back to an answer that falls between 0 and 2 pi and that's going to be 111 .8 degrees.

02:15 So that's what we'll start with for our angle.

02:17 We also need to know the value of r and we have the relationship x squared plus y squared equals r squared x is negative 2 so negative 2 squared y is 5 so 5 squared equals r squared square and add and you get 29 for r squared so r squared the square root of 29.

02:42 Okay.

02:44 So for our first answer, using an angle that falls between 0 and 2 pi, we could say that the radius is square root 29 and the angle is 111 .8 degrees.

02:57 Or in radiance, we would say the radius is square root 29 and the angle is 1 .95 radiance.

03:07 Another way to write the answer is to go 180 degrees from that angle across to the other side, and use a negative r value instead of a positive r value...

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