Question 467010
<pre>
Find the exact value of sin(2<font face="symbol">q</font>) if csc(<font face="symbol">q</font>) = -4 and 180° < <font face="symbol">q</font> < 270°

The other tutor said incorrectly that x could be positive or negative.
That's because he ignored that you said 180° < <font face="symbol">q</font> < 270°.  This pinpoints <font face="symbol">q</font> squarely in the
third quadrant, where x is clearly negative and not positive.


We use the formula sin(2<font face="symbol">q</font>) = 2sin(<font face="symbol">q</font>)cos(<font face="symbol">q</font>)

But we will need sin(<font face="symbol">q</font>) and cos(<font face="symbol">q</font>)


Let's draw the picture of angle <font face="symbol">q</font> in standard position in the
3rd quadrant, since 180° < <font face="symbol">q</font> < 270°.
Since the cosecant is the hypotenuse over the opposite and we
have 4/1, we can draw the terminal side r to be r=4 units long. 

{{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5),
     green(line(0,0,-sqrt(15),-1)), red(arc(0,0,1.5,-1.5,0,194.5)),
   locate(-2,-.5,r=4)

 )}}}


Now we draw a perpendicular from the end of the terminal side up 
to the x-axis, like this, and it will be y=-1 because the cosecant
is r/y, and since cosecant is -4 and r=-4, y=-1, and 
{{{x = -sqrt(r^2-y^2)=-sqrt(4^2-(-1)^2)=-sqrt(16-1)=-sqrt(15)}}} 

{{{drawing(400,400,-5,5,-5,5,graph(400,400,-5,5,-5,5),
     green(line(0,0,-sqrt(15),-1),line(-sqrt(15),0,-sqrt(15),-1) ), red(arc(0,0,1.5,-1.5,0,194.5)), locate(-5,-.5,y=-1),
   locate(-2,-.5,r=4),locate(-3,.6,x=-sqrt(15))
 

 )}}}
                      __
Now we see that x = -<font face = "symbol">Ö</font>
15, y = -1, and r = 4

So sin(<font face="symbol">q</font>) = {{{y/r}}} = {{{-1/4}}} and cos(<font face="symbol">q</font>) = {{{x/r}}} = {{{-sqrt(15)/4}}}

and we can substitute in

sin(2<font face="symbol">q</font>) = 2sin(<font face="symbol">q</font>)cos(<font face="symbol">q</font>)

and get

{{{sin(2theta) = 2(-1/4)(-sqrt(15)/4)= (-1/2)(-sqrt(15)/4)= sqrt(15)/8}}}

Edwin</pre>