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Hasil dari

Pertanyaan

Hasil dari fraction numerator cos space open parentheses 45 minus x close parentheses degree plus cos space open parentheses 45 plus x close parentheses degree over denominator sin space open parentheses 45 plus x close parentheses degree plus sin space open parentheses 45 minus x close parentheses degree end fraction equals.... 

  1. square root of 2 

  2. 1 half square root of 2 

  3. negative 1 

  4. negative square root of 2 

  5. 1 

Pembahasan Soal:

Dengan menggunakan rumus perkalian trigonometri, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator cos space open parentheses 45 minus x close parentheses degree plus cos space open parentheses 45 plus x close parentheses degree over denominator sin space open parentheses 45 plus x close parentheses degree plus sin space open parentheses 45 minus x close parentheses degree end fraction end cell row blank equals cell fraction numerator 2 times cos space 45 degree times up diagonal strike cos space x degree end strike over denominator 2 times sin space 45 degree times up diagonal strike cos space x degree end strike end fraction end cell row blank equals cell fraction numerator 2 times cos space 45 degree over denominator 2 times sin space 45 degree end fraction end cell row blank equals cell fraction numerator 2 times begin display style 1 half end style square root of 2 over denominator 2 times 1 half square root of 2 end fraction end cell row blank equals 1 end table   

Jadi, jawaban yang tepat adalah E.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Nur

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Terakhir diupdate 05 Juni 2021

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Pertanyaan yang serupa

Pembahasan Soal:

Rumus perkalian sinus dan kosinus adalah sebagai berikut.

open parentheses straight i close parentheses space space 2 space sin space A space cos space B equals sin space open parentheses A plus B close parentheses plus sin space open parentheses A minus B close parentheses left parenthesis ii right parenthesis space space 2 space cos space A space sin space B equals sin space open parentheses A plus B close parentheses minus sin space open parentheses A minus B close parentheses left parenthesis iii right parenthesis space 2 space cos space A space cos space B equals cos space open parentheses A plus B close parentheses plus cos space open parentheses A minus B close parentheses left parenthesis iv right parenthesis space 2 space sin space A space sin space B equals cos space open parentheses A minus B close parentheses minus cos space open parentheses A plus B close parentheses

Berdasarkan rumus no (i) dan (ii) di atas, maka:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses sin space 75 degree space cos space 15 degree close parentheses minus open parentheses cos space 75 degree space sin space 15 degree close parentheses end cell row blank equals cell 1 half open parentheses sin space open parentheses 75 degree plus 15 degree close parentheses plus sin space open parentheses 75 degree minus 15 degree close parentheses close parentheses end cell row blank blank cell negative 1 half space open parentheses sin space open parentheses 75 degree plus 15 degree close parentheses minus sin space open parentheses 75 degree minus 15 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses space sin space 90 degree plus sin space 60 degree close parentheses minus 1 half open parentheses sin space 90 degree minus sin space 60 degree close parentheses end cell row blank equals cell 1 half open parentheses 1 plus 1 half square root of 3 close parentheses minus 1 half open parentheses 1 minus 1 half square root of 3 close parentheses end cell row blank equals cell 1 half plus 1 fourth square root of 3 minus 1 half plus 1 fourth square root of 3 end cell row blank equals cell 2 over 4 square root of 3 end cell row blank equals cell 1 half square root of 3 end cell end table

Dengan demikian, hasil dari open parentheses sin space 75 degree space cos space 15 degree close parentheses minus open parentheses cos space 75 degree space sin space 15 degree close parentheses adalah 1 half square root of 3.

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Perhatikan pernyataan berikut. i) 2sin40∘cos20∘=sin30∘+sin20∘ ii) 2cos50∘sin70∘=sin120∘+sin20∘ iii) 2sin70∘sin10∘=−cos80∘+cos60∘ iv) 2cos80∘cos20∘=cos100∘+cos30∘ Pernyataan yang benar adalah ....

Pembahasan Soal:

Ingat,

Perkalian trigonometri

2sinAcosB=sin(A+B)+sin(AB)2cosAsinB=sin(A+B)sin(AB)2cosAcosB=cos(A+B)+cos(AB)2sinAsinB=cos(AB)cos(A+B)

Sudut Berelasi

sin(A)=sinA

Berdasarkan rumus tersebut, diperoleh perhitungan sebagai berikut

i) 2sin40cos20=sin30+sin20

2sin40cos20==sin(40+20)+sin(4020)sin60+sin20

Sehingga, pernyataan (i) bernilai salah

ii) 2cos50sin70=sin120+sin20

2cos50sin70====sin(50+70)sin(5070)sin120sin(20)sin120(sin20)sin120+sin20

Sehingga, pernyataan (ii) bernilai benar

iii) 2sin70sin10=cos80+cos60

2sin70sin10===cos(7010)cos(70+10)cos60cos80cos80+cos60

Sehingga, pernyataan (iii) bernilai benar

iv) 2cos80cos20=cos100+cos30

2cos80cos20==cos(80+20)+cos(8020)cos100+cos60

Sehingga, pernyataan (iv) bernilai salah

Dengan demikian, pernyataan yang benar adalah ii) dan iii)

Oleh karena itu, jawaban yang benar adalah D. 

0

Roboguru

Tanpa menggunakan tabel trigonometri, buktikan:

Pembahasan Soal:

Ingat kembali:

2 space sin space straight alpha space cos space straight beta equals sin space open parentheses straight alpha plus straight beta close parentheses plus sin space open parentheses straight alpha minus straight beta close parentheses 2 space cos space straight alpha space cos space straight beta equals cos space open parentheses straight alpha plus straight beta close parentheses minus cos space open parentheses straight alpha minus straight beta close parentheses 2 space sin space straight alpha space sin space straight beta equals negative cos space open parentheses straight alpha plus straight beta close parentheses plus cos space open parentheses straight alpha minus straight beta close parentheses sin space open parentheses 180 degree minus straight A close parentheses equals sin space straight A cos space open parentheses 90 degree plus straight A close parentheses equals negative sin space straight A 

Akan dibuktikan sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree equals 1 half. Maka:

 begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell Ruas space kiri end cell equals cell sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree end cell row blank equals cell 1 half times 2 times sin space 52 degree space sin space 68 degree minus 1 half times 2 times sin space 47 degree space cos space 77 degree minus 1 half times 2 times cos space 65 degree space cos space 81 degree end cell row blank equals cell 1 half times open parentheses 2 times sin space 52 degree space sin space 68 degree close parentheses minus 1 half times open parentheses 2 times sin space 47 degree space cos space 77 degree close parentheses minus 1 half times open parentheses 2 times cos space 65 degree space cos space 81 degree close parentheses end cell row blank equals cell 1 half open square brackets negative cos space open parentheses 52 degree plus 68 degree close parentheses plus cos space open parentheses 52 degree minus 68 degree close parentheses close square brackets minus 1 half open square brackets sin space open parentheses 47 degree plus 77 degree close parentheses plus sin space open parentheses 47 degree minus 77 degree close parentheses close square brackets minus 1 half open square brackets cos space open parentheses 65 degree plus 81 degree close parentheses plus cos space open parentheses 65 degree minus 81 degree close parentheses close square brackets end cell row blank equals cell 1 half open square brackets negative cos space 120 degree plus cos space open parentheses negative 16 degree close parentheses close square brackets minus 1 half open square brackets sin space 124 degree plus sin space open parentheses negative 30 degree close parentheses close square brackets minus 1 half open square brackets cos space 146 degree plus cos space open parentheses negative 16 degree close parentheses close square brackets end cell row blank equals cell 1 half open square brackets negative open parentheses negative 1 half close parentheses minus cos space 16 degree close square brackets minus 1 half open square brackets sin space 124 degree minus sin space 30 degree close square brackets minus 1 half open square brackets cos space 146 degree minus cos space 16 degree close square brackets end cell row blank equals cell 1 half open square brackets 1 half minus cos space 16 degree close square brackets minus 1 half open square brackets sin space 124 degree minus 1 half close square brackets minus 1 half open square brackets cos space 146 degree minus cos space 16 degree close square brackets end cell row blank equals cell 1 half open parentheses 1 half up diagonal strike negative cos space 16 degree end strike minus sin space 124 degree plus 1 half minus cos space 146 degree up diagonal strike plus cos space 16 end strike degree close parentheses end cell row blank equals cell 1 half open parentheses 1 minus sin space 124 degree minus cos space 146 degree close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses sin space 124 degree plus cos space 146 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses sin space open parentheses 180 degree minus 56 degree close parentheses plus cos space open parentheses 90 degree plus 56 degree close parentheses close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses up diagonal strike sin space 56 degree end strike up diagonal strike negative sin space 56 degree end strike close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus 0 close parentheses end cell row blank equals cell 1 half open parentheses 1 close parentheses end cell row blank equals cell 1 half end cell end table end style

Jadi, terbukti bahwa sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree equals 1 half.

0

Roboguru

Nilai dari (tan10∘+tan70∘−tan50∘) adalah ....

Pembahasan Soal:

Ingat rumus jumlah dan selisih trigonometri berikut ini:

2sinAcosB=sin(A+B)+sin(AB)

2cosAsinB=sin(A+B)sin(AB)

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

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

cosA+cosB=2cos21(A+B)cos21(AB)

Dengan menggunakan konsep di atas, diperoleh hasil:

tan10+tan70tan50=(cos10sin10+cos70sin70)cos50sin50=(cos70cos10cos70sin10+sin70cos10)cos50sin50=22(cos70cos10cos70sin10+sin70cos10)cos50sin50=(2cos70cos102cos70sin10+2sin70cos10)cos50sin50=(cos(70+10)+cos(7010)(sin(70+10)sin(7010))+(sin(70+10)+sin(7010)))cos50sin50=(cos80+cos60(sin80sin60)+(sin80+sin60))cos50sin50=cos80+212sin80cos50sin50=cos80cos50+21cos502sin80cos50sin50(cos80+21)(samakanpenyebut)=cos80cos50+21cos502sin80cos50cos80sin5021sin50=cos80cos50+21cos502sin80cos50cos80sin5021sin5022=2cos80cos50+cos5022sin80cos502cos80sin50sin50=cos(80+50)+cos(8050)+cos502(sin(80+50)+sin(8050))(sin(80+50)sin(8050))sin50=cos130+cos30+cos502(sin130+sin30)(sin130sin30)sin50=cos130+213+cos502(sin130+21)(sin13021)sin50=213+cos130+cos502sin130+1sin130+21sin50=213+cos130+cos5023+sin130sin50=213+(2cos21(130+50)cos21(13050))23+(2cos21(130+50)sin21(13050))=213+(2cos21(180)cos21(80))23+(2cos21(180)sin21(80))=213+(2cos90cos40)23+(2cos90sin40)=213+(20cos40)23+(20sin40)=213+023+0=21323=23÷23=23×32=33=33×33=333=3

Jadi, tan10+tan70tan50=3.

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Pembahasan Soal:

Sifat penjumlahan dan pengurangan trigonometri :

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

Sifat perkalian trigonometri :

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

Dengan menggunakan sifat tersebut, maka :

tan space 75 degree plus tan space 15 degree equals fraction numerator sin space 75 degree over denominator cos space 75 degree end fraction plus fraction numerator sin space 15 degree over denominator cos space 15 degree end fraction equals fraction numerator sin space 75 degree cos space 15 degree plus sin space 15 degree cos space 75 degree over denominator cos space 75 degree cos space 15 degree end fraction equals fraction numerator 1 half open parentheses 2 sin space 75 degree cos space 15 degree close parentheses plus 1 half open parentheses 2 cos space 75 degree sin space 15 degree close parentheses over denominator 1 half open parentheses 2 cos space 75 degree cos space 15 degree close parentheses end fraction equals fraction numerator 1 half open parentheses sin space 90 degree plus sin space 60 degree close parentheses plus 1 half open parentheses sin space 90 degree minus sign sin space 60 degree close parentheses over denominator 1 half open parentheses cos space 90 degree plus cos space 60 degree close parentheses end fraction equals fraction numerator 1 half open parentheses 1 plus 1 half square root of 3 close parentheses plus 1 half open parentheses 1 minus sign 1 half square root of 3 close parentheses over denominator 1 half open parentheses 0 plus 1 half close parentheses end fraction equals fraction numerator 1 half plus 1 fourth square root of 3 plus 1 half minus sign 1 fourth square root of 3 over denominator 1 fourth end fraction equals fraction numerator 1 half plus 1 half plus 1 fourth square root of 3 minus sign 1 fourth square root of 3 over denominator 1 fourth end fraction equals fraction numerator 1 over denominator 1 fourth end fraction equals 4 

Maka, tan space 75 degree plus tan space 15 degree equals 4

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

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