Question

Suppose (x, y) =(4, -5) and θ is an angle in standard position with (x, y) on its terminal side. What is the degree measure of angle θ? im not sure how to do this one problem plz show step by step

Answer 1

actually could i take the arctan of -5/4? would that give me the degree?

Answer 2

arctan (-5/4) = appr -51.34°
which would be a "clockwise" rotation and place you in quadrant IV
and stating it as a positive rotation would be 308.66°

In this case you answer is correct, but we lucked into it

Let me explain:
suppose your point had been (-4,5) which would be an angle in the second quadrant.
Your calculations of arctan (5/-4) would give us the same -51.34° , but it would not be correct.

Calculators have been programmed to give the closest angle to zero when doing any arc(trigfunction)
Since there are two answers for any arc(trigfunction) , the calculator cannot know which angle you want, and you have to establish that yourself using the good ol' CAST rule.

What I do is to use the positive fraction in my arctan, knowing from the point where the angle is
e.g. what angle does the terminal arm ending at (-4,5) make with the x-axis?

arctan (+4/5) = 51.34°
but I know I am in quadrant II, so the actual angle is
180-51.34 or 128.66°

In general, after you find the angle Ø in standard position (quad I ),

for
Quadrant I : your angle is Ø
Quadrant II : your angle is 180-Ø
Quadrant III: your angle is 180+Ø
Quadrant IV : your angle is 360-Ø