Question 1198721


Given: 

 {{{cos( theta) = -1/5}}} and {{{tan (theta) < 0}}}


find:

 {{{cot (theta)}}} and {{{sin (theta)}}}

Start with the identity:

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

Substitute {{{cos( theta) = -1/5}}}

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

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

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

{{{sin(theta)}}}=±{{{sqrt(4/5)}}}

{{{sin(theta)}}}=±{{{(2sqrt(5))/5}}}

 {{{tan (theta) < 0}}}=>{{{sin(theta)=-(2sqrt(5))/5}}}

then

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


{{{cot (theta)=cos(theta)/sin(theta)}}}...........since {{{tan (theta)<0 }}} and {{{sin(theta)=-(2sqrt(5))/5}}}, use positive solution for cos

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


so,

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

{{{cot (theta)=-1/2}}}