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Jika A dan B adalah sudut lancip yang memenuhi B−A=31​π dan sinA=31​sinB, nilai dari cos(A+B)=....

Pertanyaan

Jika A dan B adalah sudut lancip yang memenuhi begin mathsize 14px style B minus A equals 1 third pi end style dan begin mathsize 14px style sin invisible function application A equals 1 third sin invisible function application B end style, nilai dari begin mathsize 14px style cos left parenthesis A plus B right parenthesis equals end style....

  1. begin mathsize 14px style negative 1 over 7 square root of 3 end style 

  2. begin mathsize 14px style 1 over 7 square root of 3 end style 

  3. begin mathsize 14px style negative 1 over 7 end style 

  4. begin mathsize 14px style 1 over 7 end style 

  5. begin mathsize 14px style negative 1 over 21 square root of 3 end style 

M. Robo

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Karena begin mathsize 14px style B minus A equals 1 third pi end style, maka begin mathsize 14px style B equals 1 third pi plus A end style 

Karena begin mathsize 14px style sin invisible function application A equals 1 third sin invisible function application B end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin invisible function application A end cell equals cell 1 third sin invisible function application B end cell row cell sin invisible function application A end cell equals cell 1 third sin invisible function application open parentheses 1 third pi plus A close parentheses end cell row cell sin invisible function application A end cell equals cell 1 third open parentheses sin invisible function application 1 third pi cos invisible function application A plus cos invisible function application 1 third pi sin invisible function application A close parentheses end cell row cell P e r h a t i k a n space b a h w a 1 third pi end cell equals cell 1 third times 180 degree equals 60 degree. space M a k a end cell row cell sin invisible function application A end cell equals cell 1 third open parentheses sin invisible function application 1 third pi cos invisible function application A plus cos invisible function application 1 third pi sin invisible function application A close parentheses end cell row cell sin invisible function application A end cell equals cell 1 third open parentheses sin invisible function application 60 degree cos invisible function application A plus cos invisible function application 60 degree sin invisible function application A close parentheses end cell row cell sin invisible function application A end cell equals cell 1 third open parentheses 1 half square root of 3 cos invisible function application A plus 1 half sin invisible function application A close parentheses end cell row cell sin invisible function application A end cell equals cell 1 over 6 square root of 3 cos invisible function application A plus 1 over 6 sin invisible function application A end cell row cell sin invisible function application A minus 1 over 6 sin invisible function application A end cell equals cell 1 over 6 square root of 3 cos invisible function application A end cell row cell 5 over 6 sin invisible function application A end cell equals cell 1 over 6 square root of 3 cos invisible function application A end cell row cell fraction numerator sin invisible function application A over denominator cos invisible function application A end fraction end cell equals cell fraction numerator square root of 3 over denominator 5 end fraction end cell row cell tan invisible function application A end cell equals cell fraction numerator square root of 3 over denominator 5 end fraction end cell row blank blank blank end table end style 

 

Selanjutnya, yang ditanyakan adalah

undefined 

Untuk mencari nilai dari sin A dan cos A perhatikan segitiga berikut

Perhatikan bahwa diketahui A  merupakan sudut lancip. Karena undefined, maka

Karena segitiga di atas merupakan segitiga siku-siku, maka nilai x bisa didapat menggunakan teorema Pythagoras. Sehingga

begin mathsize 14px style x equals square root of 5 squared plus open parentheses square root of 3 close parentheses squared end root x equals square root of 25 plus 3 end root x equals square root of 28 end style 

Sehingga didapatkan begin mathsize 14px style sin invisible function application A equals fraction numerator square root of 3 over denominator x end fraction equals fraction numerator square root of 3 over denominator square root of 28 end fraction space d a n space cos invisible function application A equals 5 over x equals fraction numerator 5 over denominator square root of 28 end fraction end style 

Selanjutnya

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin invisible function application A end cell equals cell 1 third sin invisible function application B end cell row cell sin invisible function application B end cell equals cell 3 sin invisible function application A end cell row cell sin invisible function application B end cell equals cell 3 times fraction numerator square root of 3 over denominator square root of 28 end fraction end cell row cell sin invisible function application B end cell equals cell fraction numerator 3 square root of 3 over denominator square root of 28 end fraction end cell row blank blank blank end table end style 

Untuk mencari nilai dari cos B perhatikan segitiga berikut

Perhatikan bahwa diketahui B merupakan sudut lancip. Karena begin mathsize 14px style sin invisible function application B equals fraction numerator 3 square root of 3 over denominator square root of 28 end fraction end style, maka

Karena segitiga di atas merupakan segitiga siku-siku, maka nilai y bisa didapat menggunakan teorema Pythagoras. Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell square root of open parentheses square root of 28 close parentheses squared minus open parentheses 3 square root of 3 close parentheses squared end root end cell row y equals cell square root of 28 minus 9 times 3 end root end cell row y equals cell square root of 28 minus 27 end root end cell row y equals cell square root of 1 end cell row y equals 1 row blank blank blank row blank blank blank end table end style 

Sehingga didapatkan begin mathsize 14px style cos invisible function application B equals fraction numerator y over denominator square root of 28 end fraction equals fraction numerator 1 over denominator square root of 28 end fraction end style 

Maka didapat bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin invisible function application A end cell equals cell fraction numerator square root of 3 over denominator square root of 28 end fraction end cell row cell cos invisible function application A end cell equals cell fraction numerator 5 over denominator square root of 28 end fraction end cell row cell sin invisible function application B end cell equals cell fraction numerator 3 square root of 3 over denominator square root of 28 end fraction end cell row cell cos invisible function application B end cell equals cell fraction numerator 1 over denominator square root of 28 end fraction end cell row blank blank blank end table end style 

Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos invisible function application open parentheses A plus B close parentheses end cell equals cell cos invisible function application A cos invisible function application B minus sin invisible function application A sin invisible function application B end cell row blank equals cell fraction numerator 5 over denominator square root of 28 end fraction times fraction numerator 1 over denominator square root of 28 end fraction minus fraction numerator square root of 3 over denominator square root of 28 end fraction times fraction numerator 3 square root of 3 over denominator square root of 28 end fraction end cell row blank equals cell 5 over 28 minus 9 over 28 end cell row blank equals cell fraction numerator 5 minus 9 over denominator 28 end fraction end cell row blank equals cell negative 4 over 28 end cell row blank equals cell negative 1 over 7 end cell row blank blank blank end table end style 

Jadi, jawaban yang tepat adalah C.

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