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Jika 2 cos (x+45)∘=cos (x−45)∘, hitunglah nilai a. tan x  b. cos x  c. cos (30∘+x∘)  d. cos (x∘−30∘)

Pertanyaan

Jika 2 space cos space left parenthesis x plus 45 right parenthesis degree equals cos space left parenthesis x minus 45 right parenthesis degree comma hitunglah nilai

a. tan space x 

b. cos space x 

c. cos space left parenthesis 30 degree plus x degree right parenthesis 

d. cos space left parenthesis x degree minus 30 degree right parenthesis 

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

nilai cos space left parenthesis x degree minus 30 degree right parenthesis adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 20 square root of 30 plus 1 over 20 square root of 10 end cell end table.

Pembahasan

a. Ingat kembali rumus perbandingan trigonometri untuk penjumlahan dan selisih dua sudut berikut.

cos space open parentheses straight A plus straight B close parentheses equals cos space straight A space cos space straight B minus sin space straight A space sin space straight B
cos space open parentheses straight A minus straight B close parentheses equals cos space straight A space cos space straight B plus sin space straight A space sin space straight B  

Maka poin a dapat diselesaikan dengan perhitungan berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 space cos space left parenthesis x plus 45 right parenthesis degree end cell equals cell cos space left parenthesis x minus 45 right parenthesis degree end cell row cell 2 space cos space x space cos space 45 degree minus 2 space sin space x space sin space 45 degree end cell equals cell cos space x space cos space 45 degree plus sin space x space sin space 45 degree end cell row cell negative 2 space sin space x space sin space 45 degree minus sin space x space sin space 45 degree end cell equals cell cos space x space cos space 45 degree minus 2 space cos space x space cos space 45 degree end cell row cell negative 3 space sin space x space sin space 45 degree end cell equals cell negative cos space x space cos space 45 degree end cell row cell negative 3 space fraction numerator sin space x over denominator cos space x end fraction end cell equals cell negative fraction numerator cos space 45 degree over denominator sin space 45 degree end fraction end cell row cell negative 3 space tan space x end cell equals cell negative 1 end cell row cell tan space x end cell equals cell 1 third end cell end table end style  

 

Dengan demikian, nilai tan space x adalah 1 third.

b. Ingat kembali rumus perbandingan trigonometri berikut.

tan space x equals de over sa cos space x equals sa over mi 

Maka poin b dapat diselesaikan dengan perhitungan berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space x end cell equals cell de over sa end cell row cell 1 third end cell equals cell de over sa end cell row mi equals cell square root of 1 squared plus 3 squared end root equals square root of 10 end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space x end cell equals cell sa over mi end cell row blank equals cell fraction numerator 3 over denominator square root of 10 end fraction end cell row blank equals cell 3 over 10 square root of 10 end cell end table  

Dengan demikian, nilai cos space x adalah 3 over 10 square root of 10.

c. 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space open parentheses 30 degree plus x degree close parentheses end cell equals cell cos space 30 degree space cos space x degree minus sin space 30 degree space sin space x degree end cell row blank equals cell 1 half square root of 3 times 3 over 10 square root of 10 minus 1 half times 1 over 10 square root of 10 end cell row blank equals cell 3 over 20 square root of 30 minus 1 over 20 square root of 10 end cell end table 

Dengan demikian, nilai cos space left parenthesis 30 degree plus x degree right parenthesis adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 20 square root of 30 minus 1 over 20 square root of 10 end cell end table.

d.

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis x degree minus 30 degree right parenthesis end cell equals cell cos space x degree space cos space 30 degree plus sin space x degree space sin space 30 degree end cell row blank equals cell 3 over 10 square root of 10 times 1 half square root of 3 plus 1 over 10 square root of 10 times 1 half end cell row blank equals cell 3 over 20 square root of 30 plus 1 over 20 square root of 10 end cell end table 

Dengan demikian, nilai cos space left parenthesis x degree minus 30 degree right parenthesis adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 20 square root of 30 plus 1 over 20 square root of 10 end cell end table.

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