Question 1205306

given:

{{{2cos^2(t) + cos(t) = 1}}}

Find all exact solutions on [{{{0}}}, {{{2pi}}})


{{{2cos^2(t) + cos(t) = 1}}}

{{{2cos^2(t) + cos(t) - 1=0}}}...write {{{cos(t) }}} as {{{-cos(t) +2cos(t) }}} 

{{{2cos^2(t) -cos(t)+2cos(t) - 1=0}}}...group

{{{(2cos^2(t) +2cos(t))-(cos(t) +1)=0}}}

{{{2cos(t)(cos(t) +1)-(cos(t) +1)=0}}}

{{{(cos(t) + 1) (2cos(t) - 1) = 0}}}

solutions:

{{{(cos(t) + 1) =0}}} =>{{{cos(t) =-1 }}}=>{{{t=cos^-1(-1)}}} => {{{t=pi}}}

{{{(2cos(t) - 1) = 0 }}}=> {{{cos(t) =1/2}}} =>{{{t=cos^-1(1/2)}}} => {{{t=pi/3}}}

 all exact solutions on [{{{0}}}, {{{2pi}}})

{{{t=pi}}}

{{{t=pi/3}}}

{{{t=5pi/3}}}