How Do You Calculate the Angle of a Vector in the Third Quadrant?

[Nat3]

Homework Statement

Rewrite the following vector in terms magnitude and angle: A force vector with a magnitude of 58 lb that is in the third quadrant with an x component whose magnitude is 30 lb.

The Attempt at a Solution

This is how I drew the problem:

[PLAIN]http://img535.imageshack.us/img535/4779/vecy.png

Obviously, the answer to the first question asking the vector magnitude is just 58, but I'm having trouble getting the angle.. I tried:

$cos(\theta) = \frac{-30}{58}$
$\theta = cos^{-1}(\frac{-30}{58}) = 121.15^{\circ}$

But obviously that cannot be the answer (and it isn't..). So what am I doing wrong?

 

[PeterO]

Homework Statement

Rewrite the following vector in terms magnitude and angle: A force vector with a magnitude of 58 lb that is in the third quadrant with an x component whose magnitude is 30 lb.

The Attempt at a Solution

This is how I drew the problem:

[PLAIN]http://img535.imageshack.us/img535/4779/vecy.png

Obviously, the answer to the first question asking the vector magnitude is just 58, but I'm having trouble getting the angle.. I tried:

$cos(\theta) = \frac{-30}{58}$
$\theta = cos^{-1}(\frac{-30}{58}) = 121.15^{\circ}$

But obviously that cannot be the answer (and it isn't..). So what am I doing wrong?

You are analysing a triangle to use trig to calculate the angle. triangles do not have sides with negative length!

 

[Nat3]

But the input of the arccos function is -1<x<1, right? So why would negative input be allowed in the domain if, well, it's not allowed (like your saying)?

 

[ehild]

$cos(\theta) = \frac{-30}{58}$
$\theta = cos^{-1}(\frac{-30}{58}) = 121.15^{\circ}$

But obviously that cannot be the answer (and it isn't..). So what am I doing wrong?

Theta is the angle the force encloses with the positive x axis. There are two angles between 0 and 360° with the same cosine: if one is x, the other is (360°-x).

ehild

 

[stallionx]

Homework Statement

Rewrite the following vector in terms magnitude and angle: A force vector with a magnitude of 58 lb that is in the third quadrant with an x component whose magnitude is 30 lb.

The Attempt at a Solution

This is how I drew the problem:

[PLAIN]http://img535.imageshack.us/img535/4779/vecy.png

Obviously, the answer to the first question asking the vector magnitude is just 58, but I'm having trouble getting the angle.. I tried:

$cos(\theta) = \frac{-30}{58}$
$\theta = cos^{-1}(\frac{-30}{58}) = 121.15^{\circ}$

But obviously that cannot be the answer (and it isn't..). So what am I doing wrong?

r* ( costheta i + sintheta j ), since it is in the 3rd quadrant cos turns out to be negative and sin as well.

 

[Nat3]

Theta is the angle the force encloses with the positive x axis. There are two angles between 0 and 360° with the same cosine: if one is x, the other is (360°-x).

ehild

But shouldn't $\theta$ be the angle from the negative x-axis? If I new the y-component of the vector in the picture, then I would use $tan^{-1}(y/x)$ and the angle would be with respect to the negative x-axis; I know, because I've done it before..

So confused ...

 

[vela]

When you wrote the equation $\cos \theta = -30/58$, you assumed the convention where $\theta$ is the angle measured from the +x-axis. In addition, when you solved for $\theta$, you used the arccos function, which will only give you an angle between 0 and 180 degrees, which corresponds to being above the x-axis.

Try drawing in the point at 121.15 degrees and distance 58 from the origin. You'll see it also has an x-component equal to -30. Your picture should also make clear the following idea.

If the point you're working with is below the x-axis, like in this problem, you can find the corresponding angle by subtracting the result of arccos from 360 degrees. So for this problem, you'd find the angle to the point in question is 360-121.15 = 238.85 degrees. But again, this angle will be measured from the +x-axis.

Now how can you convert that angle from being measured from the +x-axis to being measured from the -x-axis?The other approach, which is the one PeterO is alluding to, is to assume the angle is measured from the -x-axis, since that's what you're looking for, and find the equation analogous to $\cos \theta = -30/58$ but based on lengths of the sides of the right triangle.

 

[stallionx]

But shouldn't $\theta$ be the angle from the negative x-axis? If I new the y-component of the vector in the picture, then I would use $tan^{-1}(y/x)$ and the angle would be with respect to the negative x-axis; I know, because I've done it before..

So confused ...

Please re-read my previous post and always take theta the angle with the x-axis and do not forget to keep track of which quadrant you are in, for Cos and Sine sign will change in each quadrant; ie: 3rd quadrant cos<0 , sin<0

 

[Nat3]

When you wrote the equation $\cos \theta = -30/58$, you assumed the convention where $\theta$ is the angle measured from the +x-axis. In addition, when you solved for $\theta$, you used the arccos function, which will only give you an angle between 0 and 180 degrees, which corresponds to being above the x-axis.

The other approach, which is the one PeterO is alluding to, is to assume the angle is measured from the -x-axis, since that's what you're looking for, and find the equation analogous to $\cos \theta = -30/58$ but based on lengths of the sides of the right triangle.

Thanks for your explanation! I understand my mistake now. I was trying to measure the angle from the -x axis, but was including the negative sign on the x-component, which I should not have done. I guess I need to review my trig book