When tan theta is -ve why did we assume that it's in the second quad?

[Douna2nd]

The problem says
" Two forces of magnitudes 12 and 15 Newton are applied to a point and the tangesnt of the angle between them is -3/4. Find the resultant of the two forces and the measure of its angle of inclination on the first force.
In the answer it says that the theta belongs to the second quad, and I assume this is because the tan -ve but why didn't it assume it's in the fourth quad? And why doess the -ve sign belong to the 4 on the x-axis not the 3 on the y-axis??

 

[tiny-tim]

Hi Douna2nd!

" … the tangesnt of the angle between them is -3/4"

In the answer it says that the theta belongs to the second quad, and I assume this is because the tan -ve but why didn't it assume it's in the fourth quad?

Because the convention is we always take the "principal value" for arctan, and that's between 0 and π (ie, 1st or 2nd quadrant).

And why doess the -ve sign belong to the 4 on the x-axis not the 3 on the y-axis??

Not following you.

 

[LCKurtz]

And why doess the -ve sign belong to the 4 on the x-axis not the 3 on the y-axis??

Not following you.

That's because -ve is baby-talk for "negative".

 

[HallsofIvy]

Two rays, starting at the same point, create two angles, one less than or equal to 180 degrees, the other larger than or equal to (and the two angles add to 360 degrees). The angle created is, by definition, the smaller of the two.

 

[CellCoree]

I would like to clarify that the assumption of the angle being in the second quadrant is based on the convention of using the unit circle to represent angles in mathematics. In this convention, the positive x-axis is considered to be the 0 degree angle, and the positive y-axis is considered to be the 90 degree angle. As we move counterclockwise around the unit circle, the angles increase and are labeled as 90, 180, 270, and 360 degrees.

When we have a negative tangent value, it means that the angle is between 180 and 270 degrees, which falls in the second quadrant. This is because the tangent function is defined as the ratio of the opposite side to the adjacent side of a right triangle, and in the second quadrant, both of these sides are negative.

In this specific problem, the negative tangent value of -3/4 indicates that the angle is in the second quadrant and not the fourth quadrant. This is because in the fourth quadrant, both the x and y values would be positive, resulting in a positive tangent value.

Therefore, the assumption of the angle being in the second quadrant is not based solely on the negative sign of the tangent value, but also on the convention of representing angles in the unit circle.