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Nilai dari cos105∘−cos15∘sin105∘−sin15∘​ = ...

Pertanyaan

Nilai dari begin mathsize 12px style fraction numerator sin 105 degree minus sin 15 degree over denominator cos 105 degree minus cos 15 degree end fraction end style = ...

  1. begin mathsize 12px style square root of 3 end style

  2. begin mathsize 12px style 1 third square root of 3 end style

  3. begin mathsize 12px style negative 1 third square root of 3 end style

  4. -1

  5. begin mathsize 12px style negative square root of 3 end style

Pembahasan Soal:

Ingat!

begin mathsize 12px style sin alpha minus sin beta equals 2 cos fraction numerator open parentheses alpha plus beta close parentheses over denominator 2 end fraction times sin fraction numerator open parentheses alpha minus beta close parentheses over denominator 2 end fraction  cos alpha minus cos beta equals negative 2 sin fraction numerator open parentheses alpha plus beta close parentheses over denominator 2 end fraction times sin fraction numerator open parentheses alpha minus beta close parentheses over denominator 2 end fraction  fraction numerator sin 105 degree minus sin 15 degree over denominator cos 105 degree minus cos 15 degree end fraction  equals fraction numerator 2 cos begin display style fraction numerator open parentheses 105 degree plus 15 degree close parentheses over denominator 2 end fraction end style times sin begin display style fraction numerator open parentheses 105 degree minus 15 degree close parentheses over denominator 2 end fraction end style over denominator negative 2 sin begin display style fraction numerator open parentheses 105 degree minus 15 degree close parentheses over denominator 2 end fraction end style times sin begin display style fraction numerator open parentheses 105 degree minus 15 degree close parentheses over denominator 2 end fraction end style end fraction  equals fraction numerator 2 times begin display style 1 half end style times begin display style 1 half end style times square root of 2 over denominator negative 2 times begin display style 1 half end style square root of 3 times begin display style 1 half end style square root of 2 end fraction  equals fraction numerator begin display style 1 half end style square root of 2 over denominator negative begin display style 1 half end style square root of 6 end fraction  equals 1 third square root of 3 end style

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. Munawaroh

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 05 Oktober 2021

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Pertanyaan yang serupa

Nilai dari cos75o−cos15osin105o−sin15o​=...

Pembahasan Soal:

R u m u s space p e n g u r a n g a n space sin u s space d a n space cos i n u s space a d a l a h colon  sin alpha minus sin beta equals 2 cos fraction numerator alpha plus beta over denominator 2 end fraction sin fraction numerator alpha minus beta over denominator 2 end fraction  cos alpha minus cos beta equals negative 2 sin fraction numerator alpha plus beta over denominator 2 end fraction sin fraction numerator alpha minus beta over denominator 2 end fraction  S e h i n g g a comma  fraction numerator sin 105 to the power of o minus sin 15 to the power of o over denominator cos 75 to the power of o minus cos 15 to the power of o end fraction equals fraction numerator 2 cos fraction numerator 105 plus 15 over denominator 2 end fraction sin fraction numerator 105 minus 15 over denominator 2 end fraction over denominator negative 2 sin fraction numerator 75 plus 15 over denominator 2 end fraction sin fraction numerator 75 minus 15 over denominator 2 end fraction end fraction  equals fraction numerator 2 cos 60 to the power of o sin 45 to the power of o over denominator negative 2 sin 45 to the power of o sin 30 to the power of o end fraction  equals fraction numerator 2 cross times 1 half cross times 1 half square root of 2 over denominator negative 2 cross times 1 half square root of 2 cross times 1 half end fraction  equals negative 1

0

Roboguru

Tanpa menggunakan tabel matematika maupun kalkulator, hitunglah setiap bentuk berikut. sin70∘−sin15∘cos70∘−cos15∘​

Pembahasan Soal:

Ingat rumus jumlah dan selisih trigonometri berikut ini:

cosAcosB=2sin21(A+B)sin21(AB)

sinAsinB=2cos21(A+B)sin21(AB)

Dengan menggunakan konsep di atas, diperoleh hasil:

sin70sin15cos70cos15====2cos21(70+15)sin21(7015)2sin21(70+15)sin21(7015)cos21(85)sin21(55)sin21(85)sin21(55)cos21(85)sin21(85)tan(285)

Jadi, sin70sin15cos70cos15=tan(285).

0

Roboguru

Nilai dari cos340∘−cos80∘sin280∘−sin20∘​ adalah ...

Pembahasan Soal:

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space alpha minus sin space beta end cell equals cell 2 cos space 1 half open parentheses alpha plus beta close parentheses times sin space 1 half open parentheses alpha minus beta close parentheses end cell row cell cos space alpha minus cos space beta end cell equals cell negative 2 sin space 1 half open parentheses alpha plus beta close parentheses times sin space 1 half open parentheses alpha minus beta close parentheses end cell end table 

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin space 280 degree minus sin space 20 degree over denominator cos space 340 degree minus cos space 80 degree end fraction end cell row blank equals cell fraction numerator 2 cos space 1 half open parentheses 280 degree plus 20 degree close parentheses times sin space 1 half open parentheses 280 degree minus 20 degree close parentheses over denominator negative 2 sin space 1 half open parentheses 340 degree plus 80 degree close parentheses times sin space 1 half open parentheses 340 degree minus 80 degree close parentheses end fraction end cell row blank equals cell fraction numerator 2 cos space begin display style fraction numerator 300 degree over denominator 2 end fraction end style times sin space begin display style fraction numerator 260 degree over denominator 2 end fraction end style over denominator negative 2 sin space begin display style fraction numerator 420 degree over denominator 2 end fraction end style times sin space begin display style fraction numerator 260 degree over denominator 2 end fraction end style end fraction end cell row blank equals cell negative fraction numerator cos space begin display style 150 degree end style over denominator sin space begin display style 210 degree end style end fraction end cell row blank equals cell negative fraction numerator cos space open parentheses 180 degree minus 30 degree close parentheses over denominator sin space begin display style open parentheses 180 degree plus 30 degree close parentheses end style end fraction end cell row blank equals cell negative fraction numerator negative cos space begin display style 30 degree end style over denominator negative sin space begin display style 30 degree end style end fraction end cell row blank equals cell negative fraction numerator cos space begin display style 30 degree end style over denominator sin space begin display style 30 degree end style end fraction end cell row blank equals cell negative fraction numerator begin display style 1 half square root of 3 end style over denominator begin display style 1 half end style end fraction end cell row blank equals cell negative square root of 3 end cell end table  

Jadi, table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin space 280 degree minus sin space 20 degree over denominator cos space 340 degree minus cos space 80 degree end fraction end cell 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 minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table.

0

Roboguru

tanα=34​ dengan π<α<23π​, maka cosα−sinα=...

Pembahasan Soal:

Ingat straight pi less than straight alpha less than fraction numerator 3 straight pi over denominator 2 end fractionberada pada kuadran space III, sehingga sin space dan space cos bernilai negatif. Sehingga dengan menggunakan konsep perbandingan sisi pada segitiga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space alpha end cell equals cell de over sa equals 4 over 3 end cell row mi equals cell square root of 4 squared plus 3 squared end root end cell row mi equals cell square root of 25 end cell row mi equals 5 row blank blank blank end table

Jadi,

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space straight alpha minus sin space straight alpha end cell equals cell sa over mi minus de over mi end cell row blank equals cell 3 over 5 minus 4 over 5 end cell row blank equals cell fraction numerator negative 1 over denominator 5 end fraction end cell end table


Oleh karena itu, jawaban yang benar adalah E.

0

Roboguru

cos70∘−cos50∘sin130∘−sin110∘​=...

Pembahasan Soal:

Untuk menyelesaikan soal terebut, kita dapat menggunakan rumus jumlah dan selisih trigonometri:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space A minus cos space B end cell equals cell negative 2 space sin space 1 half open parentheses A plus B close parentheses space sin space 1 half open parentheses A minus B close parentheses end cell row cell space sin space A minus sin space B end cell equals cell 2 space cos space 1 half open parentheses A plus B close parentheses space sin space 1 half open parentheses A minus B close parentheses end cell end table

Tabel sudut istimewa:

Pembahasan:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space 130 degree minus sin space 110 degree over denominator cos space 70 degree minus space cos space 50 degree end fraction end cell equals cell space fraction numerator up diagonal strike 2 space cos space 1 half open parentheses 130 degree plus 110 degree close parentheses space sin space 1 half open parentheses 130 degree minus 110 degree close parentheses over denominator up diagonal strike negative 2 end strike space sin space 1 half open parentheses 70 degree plus 50 degree close parentheses space sin space 1 half open parentheses 70 degree minus 50 degree close parentheses end fraction end cell row blank equals cell fraction numerator negative space cos space 120 degree up diagonal strike space sin space 10 degree end strike over denominator sin space 60 degree space up diagonal strike sin space 10 degree end strike end fraction end cell row blank equals cell fraction numerator negative open parentheses negative begin display style 1 half end style close parentheses over denominator begin display style 1 half end style square root of 3 end fraction end cell row blank equals cell square root of 3 end cell end table end style

Oleh karena itu, jawaban yang benar adalah A.

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