Question 1204873


given:

{{{cos(theta)= -2/3}}}, quadrant II

since {{{cos(theta)= adj/hyp}}}, we know that

{{{adj=-2}}}
{{{hyp=3}}}

then
{{{opp=sqrt(3^2-(-2)^2)}}}
{{{opp=sqrt(9-4)}}}
{{{opp=sqrt(5)}}}


then


{{{sin(theta)=opp/hyp}}}

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


In Quadrant II, ⁡ {{{sin(theta)}}} is positive, so we need {{{sin(theta)=sqrt(5)/3}}}


{{{tan(theta)=sin(theta)/cos(theta)}}}

{{{tan(theta)=(sqrt(5)/3)/(-2/3)}}}

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


{{{cot(theta)=cos(theta)/sin(theta)}}}

{{{cot(theta)=(-2/3)/(sqrt(5)/3)}}}

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

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



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

{{{sec(theta)=1/(-2/3)}}}

{{{sec(theta)=-3/2}}}



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

{{{csc(theta)=1/(sqrt(5)/3)}}}

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

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