Question 1204736


If
{{{sin(x) =1/5}}}

and {{{x }}}is in quadrant I, find the exact values of the expressions without solving for {{{x}}}

in quadrant I sin, cos, and tan are positive


{{{sin(x) =1/5}}}=>{{{a/c=1/5}}} => {{{a=1 }}}and {{{c=5}}}


using Pythagorean theorem,

{{{b=sqrt(5^2-1^2)}}}

{{{b=sqrt(24)}}}

{{{b=sqrt(4*6)}}}

{{{b=2sqrt(6)}}} or {{{b=-2sqrt(6)}}}

we need positive value

{{{b=2sqrt(6)}}}


then {{{cos(x)=(2sqrt(6))/5}}}



(a) 

{{{sin(2x)=2cos(x)* sin(x)}}}

{{{sin(2x)=2((2sqrt(6))/5) *(1/5)}}}

{{{sin(2x)=(4sqrt(6))/25}}}


(b) 

{{{cos(2x)=cos^2(x) - sin^2(x)}}}

{{{cos(2x)=((2sqrt(6))/5)^2 - (1/5)^2}}}

{{{cos(2x)=24/25- 1/25}}}

{{{cos(2x)=23/25}}}


(c) 

{{{tan(2x)=sin(2x)/cos(2x)}}}

{{{tan(2x)=((4sqrt(6))/25)/(23/25)}}}

{{{tan(2x)=(4sqrt(6))/23}}}