Question

Trig question: How to find theta, given X and Y when X and Y are both negative

Answer 1

For reference

Answer 2

mental gymnasitics ... the trig functions give us an angle in relation to the x axis so we have to make adjustments

Answer 3

if given an x and y, it tends to be tan related ..

Answer 4

CLARIFICATION:
if Y is positive, we can use arccos(x/radius) to find theta,
if X is positive, we can use arcsin(y/radius) to find theta
Is there an easy way like this to find theta if both are negative? (preferably in very few steps, to make it easy for a computer program to calculate quickly)

Answer 5

arctan( ) has range -pi/2 to pi/2
which means to find theta in the 1st and 4th quadrant we can just do arctan( y/x)+2pi*n
I put the +2pi*n depending on the domain of theta
n is an integer


now the tricky part is what to do when we get to the 2nd and 3rd quadrant
if we try doing arctan(y/x) for these two quadrants
y/x will be positive if we are in the third quadrant which means arctan(y/x) will really being giving us outputs from the 1st quadrant
so we can add pi or even pi+2pi*n or factoring out the pi ... pi(1+2n)
something similar happens for the 2nd quadrant



choosing the integer n depends on the domain of theta