Question 918834
<pre>
To find the reference number.

1. Sometimes you do nothing, because the number IS the reference number
2. Sometimes you subtract from {{{pi}}}
3. Sometimes you subtract {{{pi}}} from the number
4. Sometimes you subtract the number from {{{2pi}}}
5. Sometimes you have to subtract {{{2pi}}} 1 or more time and then do one of
    1,2,3,or 4
5. When the number is negative sometimes you just change the sign of the number.
6. Sometimes you subtract the negative number from {{{-pi}}}
7. Sometimes you subtract {{{-pi}}} from the number.
8. Sometimes you subtract the number from {{{-2pi}}}
etc. etc. etc.  

To find the reference number, you MUST draw the arc on the unit circle because
it's different for every quadrant and direction of rotation.  if it is more than
{{{2pi}}} then you must subtract {{{2pi}}} for every revolution.  You can't just
learn a bunch of rules.  There are too many.  You have to draw the arc each
time.  Positive numbers are rotated counter-clockwise and negative numbers are rotated clockwise.  

Find the reference number of t = 11&#960;/7 and t = 11&#960;/5?
t = 11&#960;/7 
t = 11&#960;/5 
<pre>
{{{expr(11/7)pi}}}{{{""=""}}}{{{expr(1&4/7)pi}}}

{{{expr(11/7)pi}}} is positive so it's the red counter-clockwise arc
around the unit circle from (1,0).  The red arc extends {{{1pi}}}
from (1,0) to (-1,0) and since it's {{{expr(1&4/7)pi}}},
it's {{{4/7}}} of that arc more, which is a tad more as the red arc we
see below.  Then the green arc is the reference number.

{{{drawing(200,200,-1.3,1.3,-1.3,1.3,graph(200,200,-1.3,1.3,-1.3,1.3),
red(arc(0,0,2,-2,0,283),arc(0,0,2.01,-2.01,0,283),arc(0,0,1.99,-1.99,0,283)),
green(arc(0,0,2,-2,283,360),arc(0,0,2.01,-2.01,283,360),arc(0,0,1.99,-1.99,283,360))

 )}}}

Since it's {{{2pi}}} units all the way around the unit circle and the
red arc is {{{expr(11/7)pi}}}, the green arc is {{{2pi-expr(11/7)pi=expr(14/7)pi-expr(11/7)pi=expr(3/7)pi=3pi/7}}} 

Answer: {{{3pi/7}}} is the reference number.

-----------

<pre>
{{{expr(11/5)pi}}}{{{""=""}}}{{{expr(2&1/5)pi}}}

{{{expr(11/5)pi}}} is positive and therefore it goes counter-clockwise.
It's also more than {{{2pi}}}, so it's more than 1 complete revolution. In 
fact it goes all the way around the unit circle and overlaps {{{1/5}}} of the way 
past (1,0) toward (-1,0). 
It's the red counter-clockwise arc below that goes around the unit circle from
(1,0) past (-1,0) on around back to (1,0) and overlaps {{{1/5}}} of the way 
past where it started at (1,0). Then the green arc is the reference number.
It's the arc that equals the amount which the red arc goes past 1 revolution
or {{{2pi}}} 

{{{drawing(200,200,-1.3,1.3,-1.3,1.3,graph(200,200,-1.3,1.3,-1.3,1.3),
red(arc(0,0,2,-2,0,180),arc(0,0,2.01,-2.01,0,180),arc(0,0,1.99,-1.99,0,180),

arc(.03,0,2.03,-2.03,180,360),arc(0.03,0,2.05,-2.05,180,360),arc(0.03,0,2.04,2.04,-1.99,180,360),arc(0,0,2.07,-2.07,0,36)



), green(arc(0,0,1.93,-1.93,0,36)))

 )}}}

Since it's {{{2pi}}} units all the way around the unit circle and the
red arc is {{{expr(11/5)pi}}}, the green arc is {{{expr(11/5)pi-2pi=expr(11/5)pi-expr(10/5)pi=expr(1/5)pi=pi/5}}} 

Answer: {{{pi/5}}} is the reference number.

Edwin</pre>