Pertanyaan

1. Jika sin θ + cos θ = 3 1 untuk θ sudut lancip, maka hitunglah: c. sin 4 θ + cos 4 θ

1. Jika $sin θ + cos θ = \frac{1}{3}$ untuk $θ$ sudut lancip, maka hitunglah:

c. $sin^{4} θ + cos^{4} θ$

P. Anggrayni

Master Teacher

Jawaban terverifikasi

Jawaban

nilai dari .

nilai dari sin to the power of 4 space theta plus cos to the power of 4 space theta equals 17 over 9

Pembahasan

Untuk menyelesaikan soal tersebut, terlebih dahulu kita kuadratkan kedua ruas persamaan , sehingga diperoleh: Selanjutnya kita faktorkan persamaan trigonometri dan ingatlahidentitas trigonometri yaitu , sehingga diperoleh: Dengan demikian, nilai dari .

Untuk menyelesaikan soal tersebut, terlebih dahulu kita kuadratkan kedua ruas persamaan sin space theta plus cos space theta equals 1 third , sehingga diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses sin space theta plus cos space theta close parentheses squared end cell equals cell open parentheses 1 third close parentheses squared end cell row cell sin squared space theta plus 2 space sin space theta space cos space theta plus cos squared space theta end cell equals cell 1 over 9 end cell row cell open parentheses sin squared space theta plus cos squared space theta close parentheses plus 2 space sin space theta space cos space theta end cell equals cell 1 over 9 end cell row cell 1 plus 2 space sin space theta space cos space theta end cell equals cell 1 over 9 end cell row cell 2 space sin space theta space cos space theta end cell equals cell 1 over 9 minus 1 end cell row cell 2 space sin space theta space cos space theta end cell equals cell fraction numerator 1 minus 9 over denominator 9 end fraction end cell row cell 2 space sin space theta space cos space theta end cell equals cell negative 8 over 9 end cell row cell sin space theta space cos space theta end cell equals cell negative fraction numerator up diagonal strike 8 over denominator 9 end fraction cross times fraction numerator 1 over denominator up diagonal strike 2 end fraction end cell row cell sin space theta space cos space theta end cell equals cell negative 4 over 9 end cell end table$

Selanjutnya kita faktorkan persamaan trigonometri open parentheses sin to the power of 4 space theta plus cos to the power of 4 space theta close parentheses dan ingatlah identitas trigonometri yaitu sin squared space theta plus cos squared space theta equals 1 , sehingga diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin to the power of 4 space theta plus cos to the power of 4 space theta end cell equals cell open parentheses sin squared space theta plus cos squared space theta close parentheses squared minus 2 times sin squared space theta times cos squared space theta end cell row blank equals cell 1 squared minus 2 open parentheses sin space theta times cos space theta close parentheses squared end cell row blank equals cell 1 minus 2 open parentheses negative 4 over 9 close parentheses end cell row blank equals cell 1 plus 8 over 9 end cell row blank equals cell fraction numerator 9 plus 8 over denominator 9 end fraction end cell row blank equals cell 17 over 9 end cell end table$

Dengan demikian, nilai dari sin to the power of 4 space theta plus cos to the power of 4 space theta equals 17 over 9 .