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Pertanyaan

Tentukan sin space open parentheses alpha minus beta close parentheses dan cos space open parentheses alpha minus beta close parentheses jika sin space alpha equals 3 over 5 comma space cos space beta equals negative 7 over 25 dengan alpha dan beta sudut tumpul!

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

nilai sin space open parentheses alpha minus beta close parentheses dan cos space open parentheses alpha minus beta close parentheses adalah 3 over 5 dan negative 44 over 125.

Pembahasan

Ingat kembali:

sin space A equals fraction numerator sisi space depan space A over denominator sisi space miring end fraction cos space A equals fraction numerator sisi space samping space A over denominator sisi space miring end fraction sin space open parentheses A minus B close parentheses equals sin space A times cos space B minus cos space A times sin space B sin space open parentheses A minus B close parentheses equals cos space A times cos space B plus sin space A times sin space B 

Diketahui sin space alpha equals 3 over 5 comma space cos space beta equals negative 7 over 25. Maka:

Dari gambar diatas, dapat ditentukan:

  • nilai x 

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell square root of 5 squared minus 3 squared end root end cell row blank equals cell square root of 25 minus 9 end root end cell row blank equals cell square root of 16 end cell row blank equals 4 end table 

  • nilai y 

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell square root of 25 squared minus 7 squared end root end cell row blank equals cell square root of 625 minus 49 end root end cell row blank equals cell square root of 576 end cell row blank equals 24 end table 

alpha dan beta sudut tumpul (kuadran II)rightwards arrow sin(+), sehingga:

cos space alpha equals negative 4 over 5 sin space beta equals 24 over 25 

Dengan demikian:

  • Nilai sin space open parentheses alpha minus beta close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses alpha minus beta close parentheses end cell equals cell sin alpha times cos space beta minus cos space alpha times sin space beta end cell row blank equals cell 3 over 5 times open parentheses negative 7 over 25 close parentheses minus open parentheses negative 4 over 5 close parentheses times 24 over 25 end cell row blank equals cell negative 21 over 125 plus 96 over 125 end cell row blank equals cell 75 over 125 end cell row blank equals cell 3 over 5 end cell end table 

  • Nilai cos space open parentheses alpha minus beta close parentheses 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space open parentheses alpha minus beta close parentheses end cell equals cell cos space alpha times cos space beta plus sin space alpha times sin space beta end cell row blank equals cell open parentheses negative 4 over 5 close parentheses times open parentheses negative 7 over 25 close parentheses minus 3 over 5 times 24 over 25 end cell row blank equals cell 28 over 125 minus 72 over 125 end cell row blank equals cell negative 44 over 125 end cell end table 

Jadi, nilai sin space open parentheses alpha minus beta close parentheses dan cos space open parentheses alpha minus beta close parentheses adalah 3 over 5 dan negative 44 over 125.

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