Question 1160460
A) 
If {{{sin(A)=-1/25 }}}and angle{{{ A}}} terminates in quadrant IV, what does {{{tan(A)}}} equal?

If {{{sin(A)=-1/25 }}}


since {{{sin(A)=opposite/hypotenuse}}}=>{{{opposite=1}}} and  {{{hypotenuse=25}}}

to find  {{{tan(A)}}} , we need  {{{cos(A)=adjacent/hypotenuse}}} 

and {{{adjacent=sqrt(25^2-1^2)=sqrt(624)=4sqrt(39)}}}

{{{cos(A)=adjacent/hypotenuse}}} 

{{{cos(A)=4sqrt(39)/25}}}


then  {{{tan(A)=sin(A)/cos(A)}}}


{{{tan(A)=(-1/25)/(4sqrt(39)/25)}}}......simplify

{{{tan(A)=-1/(4sqrt(39))}}}

{{{tan(A)=-sqrt(39)/(4*39)}}}

{{{tan(A)=-sqrt(39)/156}}} in radians


{{{A=-2.29241}}}°


in quadrant IV: {{{A=360-2.29241=357.70759}}}°



B) Given {{{tan( theta)=7/24}}}, and{{{ theta}}} terminates in Quadrant III, determine the value of cos{{{ theta}}}.

{{{tan(A)=sin(A)/cos(A)=opposite/adjacent}}}=> {{{opposite=7}}} and {{{adjacent=24}}}

find hypotenuse

{{{hypotenuse=sqrt(7^2+24^2)}}}

{{{hypotenuse=sqrt(625)}}}

{{{hypotenuse}}}=±{{{25}}}


since {{{ theta}}} terminates in Quadrant III, {{{cos(A)}}} is negative
so, {{{hypotenuse=-25}}}

{{{cos(A)=24/-25}}}
 
{{{cos(A)=-24/25}}} 

since {{{tan(theta) = 7/24}}}, use your calculator to find {{{theta = 16.26}}}°
that's in the first quadrant

in the third quadrant, the angle is {{{180 + 16.26}}}° = {{{196.26}}}°

so,

{{{A=196.26}}}°



C) 

If {{{sin(theta)= 1/2}}} and {{{theta}}} terminates in Quadrant II, what is the values of {{{csc(theta)*cot(theta)}}}?


in Quadrant II: sin is positive, cos is  negative, tanis  negative

{{{sin(A)=opposite/hypotenuse}}}=>{{{opposite=1}}} and  {{{hypotenuse=2}}}

and {{{adjacent=sqrt(2^2-1^2)=sqrt(3)}}}


then {{{cos(A)=-sqrt(3)/2}}} 


since {{{csc(theta) =hypotenuse/opposite}}} we have


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


{{{cot(theta) = adjacent/opposite}}}


{{{cot(theta) = sqrt(3)/1}}}


{{{cot(theta) = sqrt(3) }}}


then

 {{{csc(theta) * cot( theta)=2sqrt(3)}}}