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Tentukan nilai dari  cos330∘×sin(−210∘)−tan(−315∘)×cot(−330∘)

Pertanyaan

Tentukan nilai dari  begin mathsize 14px style cos space 330 degree cross times sin open parentheses negative 210 degree close parentheses minus tan open parentheses negative 315 degree close parentheses cross times text cot end text open parentheses negative 330 degree close parentheses end style

Pembahasan Soal:

Menggunakan konsep sudut berelasi di kuadran III dan IV.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos space 330 to the power of degree times sin left parenthesis negative 210 to the power of degree right parenthesis minus tan open parentheses negative 315 to the power of degree close parentheses times co t open parentheses negative 330 to the power of degree close parentheses end cell row blank equals cell cos space 330 to the power of degree times open parentheses sin left parenthesis negative 210 to the power of degree right parenthesis minus tan open parentheses negative 315 to the power of degree close parentheses close parentheses times co t open parentheses negative 330 to the power of degree close parentheses end cell row blank equals cell cos open parentheses 360 to the power of degree minus 30 to the power of degree close parentheses times open parentheses negative sin open parentheses 180 to the power of degree plus 30 to the power of degree close parentheses plus tan open parentheses 360 to the power of degree minus 45 to the power of degree close parentheses close parentheses times fraction numerator 1 over denominator negative tan open parentheses 360 to the power of degree minus 30 to the power of degree close parentheses end fraction end cell row blank equals cell cos space 30 to the power of degree times open parentheses sin space 30 to the power of degree minus tan space 45 to the power of degree close parentheses times fraction numerator 1 over denominator tan space 30 to the power of degree end fraction end cell row blank equals cell fraction numerator square root of 3 over denominator 2 end fraction times open parentheses 1 half minus 1 close parentheses times open parentheses fraction numerator 3 over denominator square root of 3 end fraction close parentheses end cell row blank equals cell fraction numerator square root of 3 over denominator 2 end fraction times fraction numerator negative 1 over denominator 2 end fraction times fraction numerator 3 over denominator square root of 3 end fraction end cell row blank equals cell negative 3 over 4 end cell end table end style


Catatan:
begin mathsize 14px style sin open parentheses negative text a end text close parentheses equals negative sin space text a end text end stylebegin mathsize 14px style tan open parentheses negative text a end text close parentheses equals negative tan space text a end text end style
nilai sin di kuadran III adalah negatif
nilai tan di kuadran IV adalah negatif
nilai cos di kuadran IV adalah positif.


Jadi, nilai dari cos space 330 degree cross times sin open parentheses negative 210 degree close parentheses minus tan open parentheses negative 315 degree close parentheses cross times text cot end text open parentheses negative 330 degree close parentheses text =- end text 3 over 4.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

D. Nuryani

Mahasiswa/Alumni Universitas Padjadjaran

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Nilai dari 2sin30∘×cos120∘sin60∘×cos210∘×tan330∘​=....

Pembahasan Soal:

Dengan menerapkan rumus sudut-sudut berelasi dan nilai sudut istimewa, diperoleh perhitungan sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin space 60 degree cross times cos space 210 degree cross times tan space 330 degree over denominator 2 space sin space 30 degree cross times cos space 120 degree end fraction end cell row blank equals cell fraction numerator sin space 60 degree cross times cos space open parentheses 180 degree plus 30 degree close parentheses cross times tan space open parentheses 360 degree minus 30 degree close parentheses over denominator 2 space sin space 30 degree cross times cos space open parentheses 180 degree minus 60 degree close parentheses end fraction end cell row blank equals cell fraction numerator sin space 60 degree cross times open parentheses negative cos space 30 degree close parentheses cross times open parentheses negative tan space 30 degree close parentheses over denominator 2 space sin space 30 degree cross times open parentheses negative cos space 60 degree close parentheses end fraction end cell row blank equals cell fraction numerator up diagonal strike begin display style 1 half end style end strike square root of 3 cross times open parentheses negative up diagonal strike begin display style 1 half end style end strike square root of 3 close parentheses cross times open parentheses negative begin display style 1 third end style square root of 3 close parentheses over denominator 2 cross times up diagonal strike begin display style 1 half end style end strike cross times open parentheses negative up diagonal strike begin display style 1 half end style end strike close parentheses end fraction end cell row blank equals cell fraction numerator negative begin display style 1 third end style open parentheses 3 close parentheses over denominator negative 2 end fraction end cell row blank equals cell fraction numerator negative 1 over denominator negative 2 end fraction end cell row blank equals cell 1 half end cell end table end style 

Jadi, nilai dari fraction numerator sin space 60 degree cross times cos space 210 degree cross times tan space 330 degree over denominator 2 space sin space 30 degree cross times cos space 120 degree end fraction adalah 1 half.

Dengan demikian, jawaban yang tepat adalah B.

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Roboguru

Hitung nilai dari: sin45π​⋅tan47π​−(cos9π)2

Pembahasan Soal:

Ingat:

1. begin mathsize 14px style sin space open parentheses straight pi plus straight x close parentheses equals negative sin space x end style

2. begin mathsize 14px style tan space open parentheses 2 straight pi minus straight x close parentheses equals negative tan space x end style

3. begin mathsize 14px style cos space left parenthesis k times 2 straight pi space plus space straight x right parenthesis equals cos space straight x end style

4. begin mathsize 14px style cos space left parenthesis straight pi minus straight x right parenthesis equals negative cos space x end style

Jadi

begin mathsize 14px style sin space fraction numerator 5 straight pi over denominator 4 end fraction times tan fraction numerator 7 straight pi over denominator 4 end fraction minus left parenthesis cos space 9 straight pi right parenthesis squared space equals sin space open parentheses straight pi plus straight pi over 4 space close parentheses times space tan space open parentheses 2 straight pi minus straight pi over 4 close parentheses minus left parenthesis cos space left parenthesis 4 times 2 straight pi plus straight pi right parenthesis right parenthesis squared space equals open parentheses negative sin space straight pi over 4 close parentheses space times space open parentheses negative tan space straight pi over 4 close parentheses minus left parenthesis cos space straight pi right parenthesis squared space equals open parentheses negative 1 half square root of 2 close parentheses space times space left parenthesis negative 1 right parenthesis minus left parenthesis cos space left parenthesis straight pi minus 0 to the power of ring operator right parenthesis right parenthesis squared space equals 1 half square root of 2 minus left parenthesis negative cos space 0 to the power of ring operator right parenthesis squared space equals 1 half square root of 2 minus left parenthesis negative 1 right parenthesis squared space equals 1 half square root of 2 minus 1 end style

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Hasil dari tan45∘−8sin210∘cos60∘+2sin330∘ adalah ....

Pembahasan Soal:

begin mathsize 14px style tan space 45 degree minus 8 sin space 210 degree space cos space 60 degree plus 2 sin space 330 degree equals tan space 45 degree minus 8 sin space left parenthesis 180 plus 30 right parenthesis degree times cos space 60 degree plus 2 sin space left parenthesis 360 minus 30 right parenthesis degree equals tan space 45 degree minus 8 left parenthesis negative sin space 30 degree right parenthesis times cos space 60 degree plus 2 left parenthesis negative sin space 30 degree right parenthesis equals 1 minus 8 open parentheses negative 1 half close parentheses times 1 half plus 2 open parentheses negative 1 half close parentheses equals 1 plus 2 minus 1 equals 2 end style

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Roboguru

Buktikan bahwa b. cotg(180−p)⋅cotg(90−p)cos(180+p)⋅sec(360−p)​=−1

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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Roboguru

Nilai dari sin150∘+4⋅cos330∘−tg225∘

Pembahasan Soal:

Nilai sinus pada sudut berelasi di kuadran II, yaitu

sin space open parentheses 180 degree minus alpha close parentheses equals sin space alpha

Nilai tangen pada sudut berelasi di kuadran III, yaitu

tan space open parentheses 180 degree plus alpha close parentheses equals tan space alpha

Nilai cosinus pada sudut berelasi di kuadran IV, yaitu

cos space open parentheses 360 degree minus alpha close parentheses equals cos space alpha

Berdasarkan konsep di atas, dapat ditentukan nilai trigonometri dari berbagai kuadran berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 150 degree end cell equals cell sin space open parentheses 180 degree minus 30 degree close parentheses end cell row blank equals cell sin space 30 degree end cell row blank equals cell 1 half end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 330 degree end cell equals cell cos space open parentheses 360 degree minus 30 degree close parentheses end cell row blank equals cell cos space 30 degree end cell row blank equals cell 1 half square root of 3 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell t g space 225 degree end cell equals cell t g space open parentheses 180 degree plus 45 degree close parentheses end cell row blank equals cell t g space 45 degree end cell row blank equals 1 end table

Hasil dari operasi hitung nilai trigonometri tersebut adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin space 150 degree plus 4 times cos space 330 degree minus t g space 225 degree end cell row blank equals cell 1 half plus 4 times 1 half square root of 3 minus 1 end cell row blank equals cell 2 square root of 3 minus 1 half end cell end table

Dengan demikian, nilai table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 150 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 330 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank t end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 225 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table 

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