How do you know for sure what quadrant?

[mathzeroh]

hello all! how do you know for sure which quadrant "they" want you to have your measure in?

for example:

Write each equation in normal form. Then find p, the measure of its normal, and "phi" the angle the normal makes with the positive x-axis.

21.-10x+5=-5y
i've got all the other stuff, it's just that when it comes to the angle measure of "phi," i get sonfused. I don't know how to recognize in which quadrant it should be. for this, I thought that it was this measure:
-26.57...but the correct answer was 333 degrees, approximately.

i know that they got this by adding 360 to -26 degrees, but WHY I don't know.

thanks in advance for any help

 

[Janitor]

For the original line, the rise over run is 2/1. I'm sure you got that far.

The angle that such a line makes to the horizontal axis is arctan(2).

The angle the normal to that line makes to the horizontal axis is arctan(2) - 90, and it is pointing into quadrant IV, so it can be thought of as a negative angle. That gives you -26.56 degrees, or so says my calculator.

Looking at it as an angle swung counterclockwise (the positive direction of rotation in the plane, by convention) from a ray going horizontally to the right, the angle is 360 - 26.56 = 333.43.

 

[M98Ranger]

!

Determining the quadrant for an angle or normal can be tricky, but there are a few key things to keep in mind. First, the x-axis is considered the reference line for angles and normals. This means that any angle or normal is measured from the positive x-axis, which is typically drawn pointing to the right.

To determine which quadrant an angle or normal is in, you need to look at the signs of the coordinates. In the example given, the equation is in the form of -10x+5=-5y, which can be rewritten as y = (-10/5)x + 1.

Since the coefficient of x is negative, the slope of the line is negative. This means that the line will be in either the second or fourth quadrant. To determine which one, you need to look at the y-intercept, which is 1 in this case.

In the second quadrant, both x and y coordinates are negative, so the angle or normal would have a negative slope and a positive y-intercept. In the fourth quadrant, both x and y coordinates are positive, so the angle or normal would have a negative slope and a negative y-intercept.

In this case, the y-intercept is positive, so the angle or normal is in the second quadrant. Now, to find the measure of phi, you need to find the reference angle first. This is the angle formed between the line and the x-axis, measured in a counterclockwise direction.

To find the reference angle, you can use the inverse tangent function (tan^-1) on a calculator. In this case, the reference angle is approximately 26.57 degrees.

To find the actual angle, you need to add or subtract 360 degrees depending on which quadrant the angle is in. In the second quadrant, you need to subtract 360 degrees from the reference angle to get 333 degrees.

In summary, to determine the quadrant for an angle or normal, you need to look at the signs of the coordinates and use the reference line of the x-axis. To find the measure of phi, you first find the reference angle and then add or subtract 360 degrees depending on the quadrant. With practice, you will become more familiar with recognizing the quadrant and finding the correct angle measure.