Question 1198723


Given: 

{{{cot(theta)=-2}}} and {{{csc(theta) < 0}}}


find:

{{{ sin (theta)}}} and {{{sec( theta)}}}

Start with the identity:

{{{1+cot^2(theta)=csc^2(theta)}}}

Substitute 

{{{cot^2(theta)=(-2)^2}}}


{{{1+(-2)^2=csc^2(theta)}}}

{{{5=csc^2(theta)}}}

Substitute 

{{{csc^2(theta)=1/sin^2(theta)}}}

{{{5=1/sin^2(theta)}}}

{{{sin^2(theta)=1/5}}}

{{{sin(theta)}}}=±{{{sqrt(1/5)}}}

{{{sin(theta)}}}=±{{{sqrt(5)/5}}}

use negative solution since {{{cot(theta)}}} negative in  Q III and  Q IV



{{{sec( theta)=1/cos(theta)}}}

use {{{cos^2(theta)=1- sin^2(theta)}}}

{{{cos^2(theta)=1- 1/5}}}

{{{cos^2(theta)=4/5}}}

{{{cos(theta)=sqrt(4/5)}}}

{{{cos(theta)}}}=±{{{(2 sqrt(5))/5}}}

then

{{{sec( theta)=1/((2 sqrt(5))/5)=sqrt(5)/2}}}

or
{{{sec( theta)=1/(-(2 sqrt(5))/5)=-sqrt(5)/2}}}

since {{{sin(theta)}}} and {{{cot(theta)}}} negative in  Q III and  Q IV


then, 

{{{sec( theta)=sqrt(5)/2}}} in  Q IV

and

{{{sec( theta)=-sqrt(5)/2}}} in  Q III