Question 930358
EXPLANATION:
The reference angle is the first quadrant angle,
an angle {{{theta}}} such that
{{{0<=theta<=pi/2}}}
(for the radian-impaired, that is {{{0^o<=theta<=90^o}}} ).
It is an angle whose trigonometric functions have the same absolute value
(you may have to change the sign to a negative sign).
It is an angle whose terminal side is symmetrical,
meaning it is reflected over the y-axis for angles in the second quadrant;
it is reflected over the x-axis for angles in the fourth quadrant,
and it is symmetrical with respect to the origin for angles in the third quadrant.
{{{drawing(300,300,-.9,.95,-.9,.95,
grid(0),red(arrow(0,0,0.52,0.9)),
red(circle(0.6,0,0.02)),locate(0.6,-0.04,red(A)),
red(circle(0.4,0.693,0.02)),locate(0.4,0.69,red(B)),
red(arc(0,0,1.7,1.7,-60,0)),green(arc(0,0,0.9,0.9,-120,0)),
locate(0.05,0.1,red(O)),green(arrow(0,0,-0.52,0.9)),
green(circle(-0.4,0.693,0.02)), locate(-0.47,0.75,green(C)),
green(arrow(0,0,-0.52,-0.9)),green(arc(0,0,1.1,1.1,120,360)),
green(circle(-0.4,-0.693,0.02)), locate(-0.4,-0.69,green(D)),
green(arrow(0,0,0.52,-0.9)),green(arc(0,0,1.3,1.3,60,360)),
green(circle(0.4,-0.693,0.02)), locate(0.42,-0.6,green(E))
)}}} Angle {{{red(AOB)}}} is the reference angle for angles
{{{green(AOC)=pi-red(AOB)}}} in the second quadrant,
{{{green(AOD)=pi+red(AOB)}}} in the third quadrant, and 
{{{green(AOE)=2pi-red(AOB)}}} in the fourth quadrant.


EXAMPLE 1:
Expressing the boundaries of the quadrants as multiples of {{{pi/4}}} for your convenience, for angles between {{{0}}} and {{{2pi}}} ,
{{{0<=theta<=pi/2=2pi/4}}} is the first quadrant;
{{{2pi/4=pi/2<theta<=pi=4pi/4}}} is the second quadrant;
{{{4pi/4=pi<theta<=3pi/2=6pi/4}}} is the third quadrant,
and {{{6pi/4=3pi/2<theta<=2pi=8pi/4}}} is the fourth quadrant.
{{{drawing(300,300,-.9,.95,-.9,.95,
grid(0),arc(0,0,1.7,1.7,-90,0),locate(0.7,0.7,pi/2),
arc(0,0,0.8,0.8,-180,0),locate(-0.28,0.28,pi),
arc(0,0,1.1,1.1,90,360),locate(-0.35,-0.21,3pi/2)
)}}}
What quadrant is {{{5pi/4}}} in?
Since {{{4pi/4=pi<5pi/4<=3pi/2=6pi/4}}} , {{{5pi/4}}} is the third quadrant,
so its reference angle is {{{5pi/4-pi=pi/4}}} .


EXAMPLE 2:
What quadrant is {{{-5pi/6}}} in?
Since {{{-pi=-6pi/6<-5pi/6<-3pi/6=-pi/2}}} ,
{{{-5pi/6}}} is also in the third quadrant.
{{{-5pi/6+pi=pi/6}}} is the reference angle.