Question 340529
{{{Sec(theta)=-1.5}}} {{{pi/2<theta<pi}}}

Find the remaining 5 trigonometric ratios for <font face = "symbol">q</font>
<pre><b>
We change {{{-1.5}}} to the fraction {{{-3/2}}}

Since {{{Sec(theta)=-3/2}}} we draw <font face = "symbol">q</font> in Quadrant II
and since {{{Sec(theta)=r/x}}} we label the value of r as 3,
the numerator of {{{-3/2}}}, considering it as {{{3/(-2)}}},
and we label the value of x as the denominator -2.

{{{drawing(400,400,-3,3,-3,3, graph(400,400,-3,3,-3,3),
triangle(0,0,-2,sqrt(5),-2,0), green(locate(.5,1.2,theta)), locate(-1.5,.3,x=-2), locate(-1,1.36,r=3), green(arc(0,0,2,-2,0,131.8103149))
 )}}}

Then we calculate the value of y by the Pythagorean relation:

{{{x^2+y^2=r^2}}}
{{{(-2)^2+y^2=3^2}}}
{{{4+y^2=9}}}
{{{y^2=5}}}
{{{y=sqrt(5)}}}

So we now label y as {{{sqrt(5)}}}

{{{drawing(400,400,-3,3,-3,3, graph(400,400,-3,3,-3,3),
triangle(0,0,-2,sqrt(5),-2,0), green(locate(.5,1.2,theta)), locate(-1.5,.3,x=-2),
locate(-2.6,1.1,y=sqrt(5)), locate(-1,1.36,r=3), green(arc(0,0,2,-2,0,131.8103149))
 )}}}

Now the trig functions for <font face = "symbol">q</font> are:

{{{sin(theta)=y/r=sqrt(5)/3}}}

{{{cos(theta)=x/r=(-2)/3=-2/3}}}

{{{tan(theta)=y/x=sqrt(5)/-2=-sqrt(5)/2}}}

{{{cot(theta)=x/y=(-2)/sqrt(5)=-2/sqrt(5)=-(2sqrt(5))/(sqrt(5)sqrt(5))=-(2sqrt(5))/5}}}

{{{sec(theta)=r/x=3/(-2)=-3/2}}}{{{(given)}}}

{{{csc(theta)=r/y=3/sqrt(5)=(3sqrt(5))/(sqrt(5)sqrt(5))=(3sqrt(5))/5}}}

Edwin</pre>