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Diketahui x + y = 3 2 π ​ dan π < x < 2 3 π ​ .Jika tan x = tan ( π + y ) , maka nilai dari cos ( 2 1 ​ x + 2 y ) adalah ....

Diketahui  dan . Jika , maka nilai dari  adalah ....

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

  2. size 14px minus size 14px 1 over size 14px 2 

  3. begin mathsize 14px style 0 end style

  4. size 14px 1 over size 14px 2 

  5. begin mathsize 14px style 1 half square root of 3 end style 

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah B.

jawaban yang tepat adalah B.

Pembahasan

Perhatikan bahwa Sehingga Karena tan ⁡x = tan ⁡y, maka Karena maka x berada di kuadran III. Sehingga tan⁡x bernilai positif. Maka . Pada interval tersebut, nilai x yang memenuhi persamaan adalah . Sehingga Dengan demikian Jadi, jawaban yang tepat adalah B.

Perhatikan bahwa

begin mathsize 14px style tan invisible function application x equals tan invisible function application open parentheses pi plus y close parentheses tan invisible function application x equals tan invisible function application y end style

Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell tan invisible function application open parentheses x plus y close parentheses end cell equals cell tan invisible function application open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses end cell row cell fraction numerator tan invisible function application x plus tan invisible function application y over denominator 1 minus tan invisible function application x tan invisible function application y end fraction end cell equals cell tan invisible function application open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses end cell end table end style

Karena tan ⁡x = tan ⁡y, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator tan invisible function application x plus tan invisible function application y over denominator 1 minus tan invisible function application x tan invisible function application y end fraction end cell equals cell tan invisible function application open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses end cell row cell fraction numerator tan invisible function application x plus tan invisible function application x over denominator 1 minus tan invisible function application x tan invisible function application x end fraction end cell equals cell negative square root of 3 end cell row cell fraction numerator 2 tan invisible function application x over denominator 1 minus tan squared invisible function application x end fraction end cell equals cell negative square root of 3 end cell row cell 2 tan invisible function application x end cell equals cell negative square root of 3 open parentheses 1 minus tan squared invisible function application x close parentheses end cell row cell 2 tan invisible function application x end cell equals cell negative square root of 3 plus square root of 3 tan squared invisible function application x end cell row 0 equals cell square root of 3 tan squared invisible function application x minus 2 tan invisible function application x minus square root of 3 end cell row 0 equals cell open parentheses square root of 3 tan invisible function application x plus 1 close parentheses open parentheses tan invisible function application x minus square root of 3 close parentheses end cell row cell tan invisible function application x end cell equals cell negative fraction numerator 1 over denominator square root of 3 end fraction text  atau  end text tan invisible function application x equals square root of 3 end cell end table end style

Karena begin mathsize 14px style pi less than x less than fraction numerator 3 pi over denominator 2 end fraction end style maka x berada di kuadran III. Sehingga tan⁡x bernilai positif. Maka begin mathsize 14px style tan invisible function application x equals square root of 3 end style.

Pada interval tersebut, nilai x yang memenuhi persamaan begin mathsize 14px style tan invisible function application x equals square root of 3 end style adalah begin mathsize 14px style x equals fraction numerator 4 pi over denominator 3 end fraction end style.

Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x plus y end cell equals cell fraction numerator 2 pi over denominator 3 end fraction end cell row cell fraction numerator 4 pi over denominator 3 end fraction plus y end cell equals cell fraction numerator 2 pi over denominator 3 end fraction end cell row y equals cell fraction numerator 2 pi over denominator 3 end fraction minus fraction numerator 4 pi over denominator 3 end fraction end cell row y equals cell negative fraction numerator 2 pi over denominator 3 end fraction end cell end table end style

Dengan demikian

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos invisible function application open parentheses 1 half x plus 2 y close parentheses end cell equals cell cos invisible function application open parentheses 1 half times fraction numerator 4 pi over denominator 3 end fraction plus 2 open parentheses negative fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses end cell row blank equals cell cos invisible function application open parentheses fraction numerator 2 pi over denominator 3 end fraction minus fraction numerator 4 pi over denominator 3 end fraction close parentheses end cell row blank equals cell cos invisible function application open parentheses negative fraction numerator 2 pi over denominator 3 end fraction close parentheses end cell row blank equals cell cos invisible function application open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses end cell row blank equals cell negative 1 half end cell end table end style

Jadi, jawaban yang tepat adalah B.

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