Pertanyaan

Selesaikanlah persamaan berikut untuk 0 ∘ < x < 36 0 ∘ cos 2 x − 3 sin x ⋅ cos x + 2 sin 2 x = 0

Selesaikanlah persamaan berikut untuk $0^{\circ} < x < 36 0^{\circ}$

$cos^{2} x - 3 sin x \cdot cos x + 2 sin^{2} x = 0$

N. Puspita

Master Teacher

Jawaban terverifikasi

Pembahasan

Dengan menggunakan rumus trigonommetri, diperoleh: Kemudian misalkan, Sehingga, Kemudian, Dengan demikian, .

Dengan menggunakan rumus trigonommetri, diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos squared space x minus 3 sin space x times cos space x plus 2 sin squared space x end cell equals 0 row cell cos squared space x minus 3 over 2 times 2 sin space x times cos space x plus 2 sin squared space x end cell equals 0 row cell open parentheses 1 minus sin squared space x close parentheses minus 3 over 2 sin space 2 x plus 2 sin squared space x end cell equals 0 row cell 1 plus sin squared space x minus 3 over 2 sin space 2 x end cell equals 0 row cell 1 plus open parentheses fraction numerator 1 over denominator negative 2 end fraction close parentheses times negative 2 sin squared space x minus 3 over 2 sin space 2 x end cell equals 0 row cell 1 plus open parentheses fraction numerator 1 over denominator negative 2 end fraction close parentheses left parenthesis cos space 2 x minus 1 right parenthesis minus 3 over 2 sin space 2 x end cell equals 0 row cell negative 3 over 2 sin space 2 x minus 1 half cos space 2 x plus 3 over 2 end cell equals 0 row cell negative 3 sin space 2 x minus cos space 2 x end cell equals cell negative 3 end cell end table$

Kemudian misalkan,

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 3 sin space 2 x minus cos space 2 x end cell equals cell k cos left parenthesis 2 x plus alpha right parenthesis degree end cell row blank equals cell k cos space 2 x degree times cos space alpha degree minus k sin space 2 x degree times sin space alpha degree end cell end table

Sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell k cos space alpha degree end cell equals cell negative 1 end cell row cell k sin space alpha degree end cell equals 3 row k equals cell square root of open parentheses negative 1 close parentheses squared plus open parentheses 3 close parentheses squared end root end cell row blank equals cell square root of 1 plus 9 end root end cell row blank equals cell square root of 10 end cell row cell tan space alpha degree end cell equals cell fraction numerator 3 over denominator negative 1 end fraction space left parenthesis kuadran space II right parenthesis end cell row cell alpha degree end cell equals cell 108 comma 43 degree end cell end table$

Kemudian,

$table attributes columnalign right center left columnspacing 0px end attributes row cell k cos space left parenthesis 2 x plus alpha right parenthesis degree end cell equals cell square root of 10 cos space left parenthesis 2 x plus 108 comma 43 right parenthesis degree equals negative 3 end cell row blank left right double arrow cell cos space left parenthesis 2 x plus 108 comma 43 right parenthesis degree equals fraction numerator negative 3 over denominator square root of 10 end fraction end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis 2 x plus 108 comma 43 right parenthesis degree end cell equals cell fraction numerator negative 3 over denominator square root of 10 end fraction end cell row cell cos space left parenthesis 2 x plus 108 comma 43 right parenthesis degree end cell equals cell cos space 161 comma 57 degree end cell row cell space 2 x plus 108 comma 43 degree end cell equals cell plus-or-minus 161 comma 57 degree plus n times 360 degree end cell row cell 2 x end cell equals cell 53 comma 14 degree plus n times 360 degree end cell row x equals cell 26 comma 57 degree plus n times 180 degree end cell row cell untuk space n end cell equals cell 0 comma space 1 end cell row x equals cell 26 comma 57 degree plus 0 times 180 degree end cell row blank equals cell 26 comma 57 degree end cell row x equals cell 26 comma 57 degree plus 1 times 180 degree end cell row blank equals cell 206 comma 57 degree end cell row cell atau space 2 x end cell equals cell negative 270 degree plus n times 360 degree end cell row x equals cell negative 135 degree plus n times 180 degree end cell row n equals cell 0 comma space 1 comma space 2 end cell row x equals cell negative 135 degree plus 0 times 180 degree end cell row blank equals cell negative 135 degree end cell row x equals cell negative 135 degree plus 1 times 180 degree end cell row blank equals cell 45 degree end cell row x equals cell negative 135 degree plus 2 times 180 degree end cell row blank equals cell 225 degree end cell end table$

Dengan demikian,

table attributes columnalign right center left columnspacing 0px end attributes row HP equals cell open curly brackets 26 comma 57 degree comma space 45 degree comma space 206 comma 57 degree comma space 225 degree close curly brackets end cell end table .