Checking on the sign from the cosine results for [math]\displaystyle{ \phi '_2 }[/math]
We have the limiting range that [math]\displaystyle{ \phi }[/math] must fall within:
| [math]\displaystyle{ -\pi \le \phi '_2 \le \pi\ Radians }[/math]
|
Examining the signs of the components which make up the angle [math]\displaystyle{ \phi }[/math] in the 4 quadrants which make up the xy plane:
| [math]\displaystyle{ For\ 0 \ge \phi '_2 \ge \frac{-\pi}{2}\ Radians }[/math]
|
| px=POSITIVE
|
| py=NEGATIVE
|
| [math]\displaystyle{ For\ 0 \le \phi '_2 \le \frac{\pi}{2}\ Radians }[/math]
|
| px=POSITIVE
|
| py=POSITIVE
|
| [math]\displaystyle{ For\ \frac{-\pi}{2} \ge \phi '_2 \ge -\pi\ Radians }[/math]
|
| px=NEGATIVE
|
| py=NEGATIVE
|
| [math]\displaystyle{ For\ \frac{\pi}{2} \le \phi '_2 \le \pi\ Radians }[/math]
|
| px=NEGATIVE
|
| py=POSITIVE
|
