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Tanpa menggunakan tabel matematika matematika maupun kalkulator, tentukan nilai dari: j cos36∘

Pertanyaan

Tanpa menggunakan tabel matematika matematika maupun kalkulator, tentukan nilai dari:

j cos space 36 degree 

Pembahasan:

Ingat kembali rumus:

sin space open parentheses 90 minus a close parentheses equals cos space a

sin space 2 straight A equals 2 space sin space straight A times cos space straight A

cos cubed space straight A equals 4 cos cubed minus 3 space cos space straight A

cos squared space straight A equals 1 minus sin squared space straight A

cos space 2 straight A equals 1 minus 2 space sin squared space straight A

Ingat kembali rumus ABC,

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 comma 2 end subscript end cell equals cell fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction end cell end table

Sehingga diperoleh perhitungan:

Ingat bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight A end cell equals cell cos space open parentheses 90 minus straight A close parentheses end cell row cell sin space 36 end cell equals cell cos space open parentheses 90 minus 36 close parentheses end cell row cell sin space 36 end cell equals cell cos space 54 end cell row cell sin space 36 end cell equals cell cos space 3 times 18 end cell row cell sin space 2 times 18 end cell equals cell cos space 3 times 18 space open parentheses misal space straight A equals 18 close parentheses space end cell row cell sin space 2 straight A end cell equals cell sin space 3 straight A end cell row cell 2 space sin space straight A times cos space straight A end cell equals cell 4 space cos cubed straight A minus 3 space cos space straight A end cell row cell 2 space sin space straight A times up diagonal strike cos space straight A end strike end cell equals cell open parentheses 4 space cos squared straight A minus 3 close parentheses up diagonal strike space cos space straight A end strike end cell row cell 2 space sin space straight A end cell equals cell 4 space cos squared straight A minus 3 end cell row cell 2 space sin space straight A end cell equals cell 4 space open parentheses 1 minus sin squared straight A close parentheses minus 3 end cell row cell 2 space sin space straight A end cell equals cell 4 space open parentheses 1 minus sin squared straight A close parentheses minus 3 end cell row cell 2 space sin space straight A end cell equals cell 4 minus 4 sin squared straight A minus 3 end cell row cell 4 sin squared straight A plus 2 space sin space straight A plus 3 end cell equals cell 0 space open parentheses misalkan space x equals sin space straight A close parentheses end cell row cell 4 x squared plus 2 x plus 3 end cell equals cell 0 space comma space a equals 4 comma space b equals 2 comma space c equals 3 end cell end table 

Dengan menggunakan rumus ABC, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 comma 2 end subscript end cell equals cell fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction end cell row blank equals cell fraction numerator negative 2 plus-or-minus square root of 2 squared minus 4 open parentheses 4 close parentheses open parentheses negative 1 close parentheses end root over denominator 2 times 4 end fraction end cell row blank equals cell fraction numerator negative 2 plus-or-minus square root of 20 over denominator 8 end fraction end cell row blank equals cell fraction numerator negative 2 plus-or-minus 2 square root of 5 over denominator 8 end fraction end cell row blank equals cell fraction numerator negative 1 plus-or-minus square root of 5 over denominator 4 end fraction end cell end table

Sehingga, kita peroleh nilai x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 1 plus-or-minus square root of 5 over denominator 4 end fraction end cell end table sehingga, sin space straight A equals sin space 18 degree equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 1 plus-or-minus square root of 5 over denominator 4 end fraction end cell end table, karena 18 degree berada di kuadran I, maka bernilai positif, sehingga sin space 18 degree equals fraction numerator negative 1 plus square root of 5 over denominator 4 end fraction.

Kemudian,

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 straight A end cell equals cell 1 minus 2 space sin squared space straight A end cell row cell cos space 2 times 18 end cell equals cell 1 minus 2 open parentheses fraction numerator negative 1 plus square root of 5 over denominator 4 end fraction close parentheses squared end cell row cell cos space 36 degree end cell equals cell 1 minus 2 open parentheses fraction numerator 6 minus 2 square root of 5 over denominator 16 end fraction close parentheses end cell row blank equals cell 1 minus fraction numerator 6 minus 2 square root of 5 over denominator 8 end fraction end cell row blank equals cell 1 minus fraction numerator 3 minus square root of 5 over denominator 4 end fraction end cell row blank equals cell fraction numerator 4 minus 3 minus square root of 5 over denominator 4 end fraction end cell row blank equals cell fraction numerator 1 minus square root of 5 over denominator 4 end fraction end cell end table   

Dengan demikian, nilai cos space 36 degree adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 minus square root of 5 over denominator 4 end fraction end cell end table

Jawaban terverifikasi

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 07 September 2021

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