Hitunglah nilainya. a. b.

Pertanyaan

Hitunglah nilainya.

a. $sin space 210 degree$

b. $cos space 315 degree$

Pembahasan Soal:

Ingat:

$sin (18 0^{\circ} + a) cos (36 0^{\circ} - a) = = - sin a cos a$

$sin 21 0^{\circ} = = = sin (18 0^{\circ} + 3 0^{\circ}) - sin 3 0^{\circ} - \frac{1}{2}$

Dengan demikian, nilai $sin space 210 degree$ adalah $- \frac{1}{2}$ .

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 315 degree end cell equals cell cos open parentheses 360 degree minus 45 degree close parentheses end cell row blank equals cell cos space 45 degree end cell row blank equals cell 1 half square root of 2 end cell end table$

Dengan demikian, nilai $cos space 315 degree$ adalah $1 half square root of 2$ .

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. Bella

Mahasiswa/Alumni UIN Syarif Hidayatullah Jakarta

Terakhir diupdate 30 Agustus 2021

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Tentukan hasil dari .

Pembahasan Soal:

Ingatlah aturan perbandingan trigonometri sudut berelasi

$cos space open parentheses 180 degree plus alpha close parentheses equals negative cos space alpha$

dan

$sin space open parentheses 360 degree minus alpha close parentheses equals negative sin space alpha$ .

Maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 240 degree end cell equals cell cos space open parentheses 180 degree plus 60 degree close parentheses end cell row blank equals cell negative cos space 60 degree end cell row blank equals cell negative 1 half end cell end table$

dan

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 300 degree end cell equals cell sin space open parentheses 360 degree minus 60 degree close parentheses end cell row blank equals cell negative sin space 60 degree end cell row blank equals cell negative 1 half square root of 3 end cell end table$

Dengan demikian,

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 240 degree minus sin space 300 degree end cell equals cell negative 1 half minus open parentheses negative 1 half square root of 3 close parentheses end cell row blank equals cell negative 1 half plus 1 half square root of 3 end cell row blank equals cell 1 half open parentheses square root of 3 minus 1 close parentheses end cell end table$

Jadi, hasil dari $cos space 240 degree minus sin space 300 degree$ adalah $1 half open parentheses square root of 3 minus 1 close parentheses$ .

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nilai

Pembahasan Soal:

Ingat:

$begin mathsize 14px style sin space left parenthesis 180 degree minus A right parenthesis equals sin space A cos space left parenthesis 180 degree plus A right parenthesis equals negative cos space A cos space left parenthesis 360 degree minus A right parenthesis equals cos space A end style$

Jadi

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space 150 degree plus sin space 120 degree over denominator cos space 210 degree minus cos space 300 degree end fraction end cell equals cell fraction numerator sin space left parenthesis 180 degree minus 30 degree right parenthesis plus sin space left parenthesis 180 degree minus 60 degree right parenthesis over denominator cos space left parenthesis 180 degree plus 30 degree right parenthesis minus cos space left parenthesis 360 degree minus 60 degree right parenthesis end fraction end cell row blank equals cell fraction numerator sin space 30 degree plus sin space 60 degree over denominator negative cos space 30 degree minus cos space 60 degree end fraction end cell row blank equals cell fraction numerator begin display style 1 half plus 1 half square root of 3 end style over denominator negative begin display style 1 half end style square root of 3 minus begin display style 1 half end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half plus 1 half square root of 3 end style over denominator negative open parentheses begin display style 1 half end style square root of 3 plus begin display style 1 half end style close parentheses end fraction end cell row blank equals cell negative 1 end cell end table$

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Tentukan hasil dari .

Pembahasan Soal:

Ingat!

$cos blank open parentheses 180 degree plus alpha close parentheses equals negative cos space alpha$

$sin space left parenthesis 360 degree minus alpha right parenthesis equals negative sin space alpha$

sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell Cos space 240 degree minus Sin space 300 degree end cell equals cell Cos blank open parentheses 180 degree plus 60 degree close parentheses minus blank Sin space left parenthesis 360 degree minus 60 degree right parenthesis end cell row blank equals cell negative cos space 60 degree minus left parenthesis negative sin space 60 degree right parenthesis end cell row blank equals cell negative 1 half minus left parenthesis negative 1 half square root of 3 right parenthesis end cell row blank equals cell negative 1 half plus 1 half square root of 3 end cell row blank equals cell 1 half square root of 3 minus 1 half end cell end table$

Dengan demikian, dapat disimpulkan bahwa hasil dari $cos space 240 degree minus sin space 300 degree$ adalah $1 half square root of 3 minus 1 half$ .

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Nilai dari

Pembahasan Soal:

Ingat!

$sin left parenthesis 180 degree minus alpha right parenthesis equals sin space alpha$
$sin left parenthesis 180 degree plus alpha right parenthesis equals negative sin space alpha$
$cos space left parenthesis 180 degree minus alpha right parenthesis equals negative cos space alpha$
$cos left parenthesis 360 degree minus alpha right parenthesis equals cos space alpha$

Maka:

$begin mathsize 14px style fraction numerator sin space 120 degree plus sin space 150 degree over denominator cos space 210 degree minus cos space 300 degree end fraction equals midline horizontal ellipsis end style$ $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin space 120 degree plus sin space 150 degree over denominator cos space 210 degree minus cos space 300 degree end fraction end cell row blank equals cell fraction numerator sin left parenthesis 180 degree minus 60 degree right parenthesis plus sin left parenthesis 180 degree minus 30 degree right parenthesis over denominator cos left parenthesis 180 degree plus 30 degree right parenthesis minus cos left parenthesis 360 degree minus 60 degree right parenthesis end fraction end cell row blank equals cell fraction numerator sin space 60 degree plus sin space 30 degree over denominator negative cos space 30 degree minus cos space 60 degree end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style square root of 3 plus begin display style 1 half end style over denominator negative begin display style 1 half end style square root of 3 minus begin display style 1 half end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half square root of 3 plus 1 half end style over denominator negative open parentheses begin display style 1 half end style square root of 3 plus begin display style 1 half end style close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator negative 1 end fraction end cell row blank equals cell negative 1 end cell end table$

Jadi, $begin mathsize 14px style fraction numerator sin space 120 degree plus sin space 150 degree over denominator cos space 210 degree minus cos space 300 degree end fraction equals negative 1 end style$ .

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Buktikan bahwa b.

Pembahasan Soal:

Dengan menggunakan hubungan sudut berelasi diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator cos open parentheses 180 plus p close parentheses times sec open parentheses 360 minus p close parentheses over denominator cotg open parentheses 180 minus p close parentheses times cotg open parentheses 90 minus p close parentheses end fraction end cell equals cell fraction numerator negative cos space p times sec space p over denominator negative cotg space p times tan space p end fraction end cell row blank equals cell fraction numerator cos space p times begin display style fraction numerator 1 over denominator cos space p end fraction end style over denominator begin display style fraction numerator 1 over denominator space tan space p end fraction end style times tan space p end fraction end cell row blank equals cell 1 over 1 end cell row blank equals 1 end table$

Sehingga $fraction numerator cos open parentheses 180 plus p close parentheses times sec open parentheses 360 minus p close parentheses over denominator cotg open parentheses 180 minus p close parentheses times cotg open parentheses 90 minus p close parentheses end fraction equals 1$ .

Dengan demikian tidak terbukti bahwa $fraction numerator cos open parentheses 180 plus p close parentheses times sec open parentheses 360 minus p close parentheses over denominator cotg open parentheses 180 minus p close parentheses times cotg open parentheses 90 minus p close parentheses end fraction equals negative 1$ .

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