a. Buktikanlah: sin5x=14sin5x−18sin3x+5sinx. (Petunjuk: 5x=3x+2x) b. Gunakan jawaban (a) untuk menghitung: (i) 14sin518∘−18sin318∘+5sin18∘ (ii) 14sin56∘−18sin36∘+5sin6∘

Pertanyaan

a. Buktikanlah:

$\sin\;5x=14\;\sin^5x-18\;\sin^3x+5\;\sin\;x$ .
(Petunjuk: $5x=3x+2x$ )

b. Gunakan jawaban (a) untuk menghitung:

(i) $14\;\sin^5\;18^\circ-18\;\sin^3\;18^\circ+5\;\sin\;18^\circ$
(ii) $14\;\sin^5\;6^\circ-18\;\sin^3\;6^\circ+5\;\sin\;6^\circ$

Pembahasan Soal:

Diasumsikan bahwa soal tersebut

a. membuktikan $sin space 5 x equals 16 space sin to the power of 5 x minus 20 space sin cubed x plus 5 space sin space x$ .
b. (i) Nilai dari $16 space sin to the power of 5 space 18 degree minus 20 space sin cubed space 18 degree plus 5 space sin space 18 degree$
(ii) Nilai dari $16 space sin to the power of 5 space 6 degree minus 20 space sin cubed space 6 degree plus 5 space sin space 6 degree$ .

Ingat bahwa:

$sin left parenthesis straight A plus straight B right parenthesis equals sin space straight A space cos space straight B plus cos space straight A space sin space straight B$
$cos squared space straight A equals 1 minus sin squared space straight A$
$sin space 2 straight A equals 2 space sin space straight A space cos space straight A$
$cos space 2 straight A equals cos squared space straight A minus sin squared space straight A$

a. Membuktikan $sin space 5 x equals 16 space sin to the power of 5 x minus 20 space sin cubed x plus 5 space sin space x$ :

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 5 x end cell equals cell sin left parenthesis 3 x plus 2 x right parenthesis end cell row cell sin space 5 x end cell equals cell left parenthesis sin space 2 x space cos space x plus cos space 2 x space sin space x right parenthesis cos space 2 x end cell row blank blank cell plus left parenthesis cos space 2 x space cos space x – sin space 2 x space sin space x right parenthesis space sin space 2 x end cell row cell sin space 5 x end cell equals cell sin space 2 x space cos space 2 x space cos space x plus cos squared space 2 x space sin space x end cell row blank blank cell plus left parenthesis sin space 2 x space cos space 2 x space cos space x – sin squared space 2 x space sin space x right parenthesis end cell row cell sin space 5 x end cell equals cell 2 sin space 2 x space cos space 2 x space cos space x end cell row blank blank cell plus cos squared space 2 x space sin space x – space sin squared space 2 x space sin space x end cell row cell sin space 5 x end cell equals cell 2 left parenthesis 2 sin space x space cos space x right parenthesis left parenthesis cos squared x – sin squared x right parenthesis cos space x end cell row blank blank cell plus left parenthesis cos squared x – sin squared x right parenthesis squared sin space x – left parenthesis 2 sin space x space cos space x right parenthesis squared sin space x end cell row cell sin space 5 x end cell equals cell 4 left parenthesis sin space x space cos squared space x right parenthesis left parenthesis left square bracket 1 – sin squared x right square bracket space – sin squared x right parenthesis end cell row blank blank cell plus left parenthesis left square bracket 1 – sin squared x right square bracket – sin squared x right parenthesis squared sin space x – left parenthesis 4 sin squared space x space cos squared space x right parenthesis sin space x space end cell row cell sin space 5 x end cell equals cell 4 left parenthesis sin space x space left square bracket 1 – sin squared x right square bracket right parenthesis left parenthesis 1 – 2 sin squared x right parenthesis plus left parenthesis 1 – 2 sin squared x right parenthesis squared sin space x space – 4 sin cubed space x space left square bracket 1 – sin squared x right square bracket end cell row cell sin space 5 x end cell equals cell 4 sin space x left parenthesis 1 – sin squared x right parenthesis left parenthesis 1 – 2 sin squared x right parenthesis plus left parenthesis 1 – 4 sin squared x plus 4 sin to the power of 4 x right parenthesis sin space x end cell row blank blank cell – 4 sin cubed space x plus 4 sin to the power of 5 x end cell row cell sin space 5 x end cell equals cell left parenthesis 4 sin space x – 4 sin cubed x right parenthesis left parenthesis space 1 – 2 sin squared x right parenthesis plus sin space x – 4 sin cubed x end cell row blank blank cell plus 4 sin to the power of 5 x – 4 sin cubed space x space plus 4 sin to the power of 5 x end cell row cell sin space 5 x end cell equals cell 4 sin space x – 8 sin cubed x – 4 sin cubed x plus 8 sin to the power of 5 x end cell row blank blank cell plus sin space x – 8 sin cubed x plus 8 sin to the power of 5 x end cell row cell sin space 5 x end cell equals cell 5 sin space x minus 20 sin cubed x plus 16 sin to the power of 5 x end cell end table$

Jadi, Terbukti bahwa $sin space 5 x equals 16 space sin to the power of 5 x minus 20 space sin cubed x plus 5 space sin space x$ .

b. (i) Mencari nilai dari $16 space sin to the power of 5 space 18 degree minus 20 space sin cubed space 18 degree plus 5 space sin space 18 degree$

Ingat bahwa $sin space 5 x equals 16 space sin to the power of 5 x minus 20 space sin cubed x plus 5 space sin space x$ , sehingga:

$table attributes columnalign right center left columnspacing 0px end attributes row cell 16 space sin to the power of 5 space 18 degree minus 20 space sin cubed space 18 degree plus 5 space sin space 18 degree end cell equals cell sin space left parenthesis 5 times 18 degree right parenthesis end cell row blank equals cell sin space left parenthesis 90 degree right parenthesis end cell row blank equals 1 end table$

Jadi, Nilai dari $16 space sin to the power of 5 space 18 degree minus 20 space sin cubed space 18 degree plus 5 space sin space 18 degree$ adalah 1.

(ii) Mencari nilai dari $16 space sin to the power of 5 space 6 degree minus 20 space sin cubed space 6 degree plus 5 space sin space 6 degree$ :

Ingat bahwa $sin space 5 x equals 16 space sin to the power of 5 x minus 20 space sin cubed x plus 5 space sin space x$ , sehingga:

$table attributes columnalign right center left columnspacing 0px end attributes row cell 16 space sin to the power of 5 space 6 degree minus 20 space sin cubed space 6 degree plus 5 space sin space 6 degree end cell equals cell sin space left parenthesis 5 times 6 degree right parenthesis end cell row blank equals cell sin space left parenthesis 30 degree right parenthesis end cell row blank equals cell 1 half end cell end table$

Jadi, Nilai dari $16 space sin to the power of 5 space 6 degree minus 20 space sin cubed space 6 degree plus 5 space sin space 6 degree$ adalah $1 half$ .

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Yoga

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 06 Oktober 2021