Final answer:
The information provided suggests a mathematics problem involving trigonometry and physics concepts such as kinematics and wave mechanics, likely at the high school or early college level.
Step-by-step explanation:
This seems to be a mathematics question regarding trigonometric functions, specifically dealing with tangent and sine functions. The given information implies that the concept of quadrant analysis is also involved, as trigonometric functions can yield different signs depending on the quadrant they are in.
The reference to equations like -b ± √b² - 4ac / 2a and notes about angles measured in counterclockwise direction from the positive x-axis suggest that the student is dealing with pre-calculus or physics problems involving kinematics, where time (t) and velocities (v') are calculated. Certain terms such as m1 sin 01 -V'1 m2 sin 02 can be related to the conservation of momentum or similar principles in physics.
There is also a mention of wave equations, indicating the topic could involve an understanding of wave motion and interference patterns. Phrases like antinode and small-angle approximation suggest analytical approaches to wave behavior, potentially relevant in a physics context.
Since the student's query is not fully stated and carries typos and incomplete sentences, it is unclear what exact problem they need assistance with. However, the discussion of quadrants, trigonometric functions, waves, and kinematics is typical of high-school to early college-level mathematics or physics assignments.