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Pertanyaan

Nyatakan bentuk berikut dalam bentuk sin x dan cos x . a. sin ( x + 13 5 ∘ ) b. cos ( x + 13 5 ∘ )

Nyatakan bentuk berikut dalam bentuk  dan .

a. 

b. 

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Y. Fathoni

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta.

Jawaban terverifikasi

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diperoleh dan .

diperoleh sin space open parentheses x plus 135 degree close parentheses equals 1 half square root of 2 open parentheses cos space x minus sin space x close parentheses dan cos space open parentheses x plus 135 degree close parentheses equals negative 1 half square root of 2 open parentheses sin space x plus cos space x close parentheses.

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Pembahasan

Gunakan konsep rumus sinus dan cosinusjumlah dua sudut. Ingat kembali nilai trigonometri sudut istimewa di kuadran II. a. Perhatikan perhitungan berikut. Diperoleh nilai . b. Perhatikan perhitungan berikut. Diperoleh nilai . Jadi, diperoleh dan .

Gunakan konsep rumus sinus dan cosinus jumlah dua sudut.

table attributes columnalign right center left columnspacing 2px end attributes row cell sin space open parentheses alpha plus beta close parentheses end cell equals cell sin space alpha times cos space beta plus cos space alpha times sin space beta end cell row cell cos space open parentheses alpha plus beta close parentheses end cell equals cell cos space alpha times cos space beta minus sin space alpha times sin space beta end cell end table

Ingat kembali nilai trigonometri sudut istimewa di kuadran II.

table attributes columnalign right center left columnspacing 2px end attributes row cell sin space 135 degree end cell equals cell 1 half square root of 2 end cell row cell cos space 135 degree end cell equals cell negative 1 half square root of 2 end cell end table

a. sin space open parentheses x plus 135 degree close parentheses
Perhatikan perhitungan berikut.

table attributes columnalign right center left columnspacing 2px end attributes row cell sin space open parentheses alpha plus beta close parentheses end cell equals cell sin space alpha times cos space beta plus cos space alpha times sin space beta end cell row cell sin space open parentheses x plus 135 degree close parentheses end cell equals cell sin space x times cos space 135 degree plus cos space x times sin space 135 degree end cell row blank equals cell sin space x times open parentheses negative 1 half square root of 2 close parentheses plus cos space x times 1 half square root of 2 end cell row blank equals cell negative 1 half square root of 2 space sin space x plus 1 half square root of 2 space cos space x end cell row blank equals cell 1 half square root of 2 open parentheses negative sin space x plus cos space x close parentheses end cell row cell sin space open parentheses x plus 135 degree close parentheses end cell equals cell 1 half square root of 2 open parentheses cos space x minus sin space x close parentheses end cell end table 

Diperoleh nilai sin space open parentheses x plus 135 degree close parentheses equals 1 half square root of 2 open parentheses cos space x minus sin space x close parentheses.

b. cos space open parentheses x plus 135 degree close parentheses
Perhatikan perhitungan berikut.

table attributes columnalign right center left columnspacing 2px end attributes row cell cos space open parentheses alpha plus beta close parentheses end cell equals cell cos space alpha times cos space beta minus sin space alpha times sin space beta end cell row cell cos space open parentheses x plus 135 degree close parentheses end cell equals cell cos space x times cos space 135 degree minus sin space x times sin space 135 degree end cell row blank equals cell cos space x times open parentheses negative 1 half square root of 2 close parentheses minus sin space x times 1 half square root of 2 end cell row blank equals cell negative 1 half square root of 2 space cos space x minus 1 half square root of 2 space sin space x end cell row blank equals cell negative 1 half square root of 2 open parentheses cos space x plus sin space x close parentheses end cell row cell cos space open parentheses x plus 135 degree close parentheses end cell equals cell negative 1 half square root of 2 open parentheses sin space x plus cos space x close parentheses end cell end table

Diperoleh nilai cos space open parentheses x plus 135 degree close parentheses equals negative 1 half square root of 2 open parentheses sin space x plus cos space x close parentheses.

Jadi, diperoleh sin space open parentheses x plus 135 degree close parentheses equals 1 half square root of 2 open parentheses cos space x minus sin space x close parentheses dan cos space open parentheses x plus 135 degree close parentheses equals negative 1 half square root of 2 open parentheses sin space x plus cos space x close parentheses.

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