# Why is the arccos of a negative value an obtuse angle?

For the sake of simplicity, I will ask the question in 2D:

Why is the arccos of a negative number (between -1 and 0, not including 0) an obtuse angle measured from the +X axis by definition, and not an acute angle measured from the -X axis?

I understand how to show empirically why it is true, using the unit circle to first find the angle between the vector and the -x axis, then taking 180-angle.

But, when I try to show it through a simple drawing, I get confused.

In Image 1, I have the vector from the origin pointing to (5,5), and the angle I calculate is 45 degrees using the given equation. Using the same equation in Image 2, the equation yields a value of 135 degrees, but the drawing seems to suggest that I am finding said angle using the cosine of -5/sqrt(50), which would make it the angle between the vector and the -x axis the one that I am calculating, not the angle between the vector and the +x axis. Perhaps I am not explaining my confusion well enough, but I hope it comes through.

• The right triangle is only useful for sines and cosines between 0 and 1. Beyond that, you should forget about the triangle entirely. The angle is always with respect to the positive x-axis, and the cosine is always the projection on the x-axis. – Rahul Jul 19 '17 at 9:13
• In diagrams like that, it is conventional to measure angles anti-clockwise from the positive $x$-axis – Henry Jul 19 '17 at 9:13
• Thank you Rahul and Henry, I am beginning to understand better now. – kreeser1 Jul 20 '17 at 2:36