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Tentukan himpunan penyelesaian dari persamaan berikut untuk −360∘≤x≤360∘ cos 2x∘+cos x∘−2cos 2x∘=0

Pertanyaan

Tentukan himpunan penyelesaian dari persamaan berikut untuk negative 360 degree less or equal than x less or equal than 360 degree

cos space 2 x degree plus cos space x degree minus 2 cos space 2 x degree equals 0

Z. Apriani

Master Teacher

Mahasiswa/Alumni Institut Teknologi Bandung

Jawaban terverifikasi

Pembahasan

Dengan menggunakan rumus trigonometri, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 x degree plus cos space x degree minus 2 cos space 2 x degree end cell equals 0 row cell cos space x degree minus cos space 2 x degree end cell equals 0 row cell cos space x degree minus open parentheses 2 space cos squared x degree minus 1 close parentheses end cell equals 0 row cell 2 space cos squared x degree minus cos space x degree minus 1 end cell equals 0 row cell open parentheses 2 cos space x degree plus 1 close parentheses left parenthesis cos space x degree minus 1 right parenthesis end cell equals 0 row cell 2 cos space x degree plus 1 end cell equals 0 row cell cos space x degree end cell equals cell negative 1 half space atau end cell row cell cos space x degree minus 1 end cell equals 0 row cell cos space x degree end cell equals 1 end table

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space x degree end cell equals cell negative 1 half end cell row cell cos space x degree end cell equals cell cos space 120 degree end cell row cell x degree end cell equals cell 120 degree plus n times 360 degree end cell row n equals cell 0 comma space 1 end cell row cell x degree end cell equals cell 120 degree plus 0 times 360 degree end cell row blank equals cell 120 degree end cell row cell x degree end cell equals cell 120 degree plus 1 times 360 degree end cell row blank equals cell 480 degree end cell row blank blank atau row cell x degree end cell equals cell negative 120 degree plus n times 360 degree end cell row n equals cell 0 comma space 1 end cell row cell x degree end cell equals cell negative 120 degree plus 0 times 360 degree end cell row blank equals cell negative 120 degree end cell row cell x degree end cell equals cell negative 120 degree plus 1 times 360 degree end cell row blank equals cell 240 degree end cell row cell cos space x degree end cell equals cell cos space open parentheses negative x degree close parentheses end cell row cell cos space x degree end cell equals cell cos space open parentheses negative 120 degree close parentheses end cell row cell x degree end cell equals cell negative 120 degree end cell row cell cos space x degree end cell equals cell cos space open parentheses negative 240 degree close parentheses end cell row cell x degree end cell equals cell negative 240 degree end cell end table

atau

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space x degree end cell equals 1 row cell cos space x degree end cell equals cell cos space 0 degree end cell row cell x degree end cell equals cell 0 degree plus n times 360 degree end cell row n equals cell 0 comma space 1 end cell row cell x degree end cell equals cell 0 degree plus 0 times 360 degree end cell row blank equals cell 0 degree end cell row cell x degree end cell equals cell 0 degree plus 1 times 360 degree end cell row blank equals cell 36 0 degree end cell row cell cos space x degree end cell equals cell cos space open parentheses negative x degree close parentheses end cell row cell cos space x degree end cell equals cell cos space open parentheses negative 360 degree close parentheses end cell row cell x degree end cell equals cell negative 360 degree end cell end table

Dengan demikian,

HP equals open curly brackets negative 360 degree comma space minus 240 degree comma space minus 120 degree comma space 0 degree comma space 120 degree comma space 240 degree comma space 360 degree close curly brackets

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