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Selesaikanlah masing-masing persamaan berikut. a. tan−1x=cos−1x

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Selesaikanlah masing-masing persamaan berikut.

a. tan to the power of negative 1 end exponent x equals cos to the power of negative 1 end exponent x 

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

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penyelesaian dari persamaan trigonometri tersebut adalah x equals 0 comma 79 space atau space x equals 1.

Pembahasan

Ingat!

Pada bentuk persamaan kuadrat a x squared plus b x plus c equals 0 berlaku rumus kuadratik x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

  • Misalkan tan to the power of negative 1 end exponent x equals p maka tan space p equals x   
  • MIsalkan cos to the power of negative 1 end exponent x equals q maka cos space q equals x 
  • tan to the power of negative 1 end exponent x equals cos to the power of negative 1 end exponent x maka p equals q 

table attributes columnalign right center left columnspacing 0px end attributes row x equals x row cell tan space p end cell equals cell cos space q end cell row cell tan space q end cell equals cell cos space q end cell row cell fraction numerator sin space q over denominator cos space q end fraction end cell equals cell cos space q end cell row cell sin space q end cell equals cell cos space q. cos space q end cell row cell sin space q end cell equals cell cos squared q end cell row 0 equals cell cos squared q minus sin space q end cell row 0 equals cell left parenthesis 1 minus sin squared q minus sin space q right parenthesis end cell row 0 equals cell negative sin squared q minus sin space q plus 1 end cell row 0 equals cell sin squared q plus sin space q minus 1 end cell end table 

Melalui rumus kuadratik persamaan kuadrat, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space q end cell equals cell fraction numerator negative 1 plus-or-minus square root of 1 squared minus 4 left parenthesis 1 right parenthesis left parenthesis negative 1 right parenthesis end root over denominator 2 left parenthesis 1 right parenthesis end fraction end cell row blank equals cell fraction numerator negative 1 plus square root of 5 over denominator 2 end fraction end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank q end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table 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 2 end fraction end cell end table 

sin space q equals fraction numerator negative 1 plus square root of 5 over denominator 2 end fraction space atau space sin space q equals fraction numerator negative 1 minus square root of 5 over denominator 2 end fraction 

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space q end cell equals cell fraction numerator negative 1 plus square root of 5 over denominator 2 end fraction end cell row cell sin space q end cell equals cell sin space 38 comma 17 degree end cell row q equals cell 38 comma 17 degree plus k.360 degree end cell row blank blank blank row cell sin space q end cell equals cell fraction numerator negative 1 minus square root of 5 over denominator 2 end fraction end cell row cell sin space q end cell equals cell sin space left parenthesis negative 1 comma 57 degree right parenthesis end cell row q equals cell negative 1 comma 57 degree plus k.360 degree end cell end table

Sehingga: 

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell cos space q end cell row blank equals cell cos space 38 comma 17 degree end cell row blank equals cell 0 comma 79 end cell row blank blank blank row x equals cell cos left parenthesis negative 1 comma 57 degree right parenthesis end cell row blank equals 1 end table  

Jadi, penyelesaian dari persamaan trigonometri tersebut adalah x equals 0 comma 79 space atau space x equals 1.

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