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Bentuk paling sederhana dari  adalah ....

Pertanyaan

Bentuk paling sederhana dari sec space alpha minus sin space alpha space tan space alpha adalah ....

Pembahasan Soal:

Perhatikan rumus trigonometri berikut!

table attributes columnalign right center left columnspacing 0px end attributes row cell sec space alpha end cell equals cell fraction numerator 1 over denominator cos space alpha end fraction end cell row cell tan space alpha end cell equals cell fraction numerator sin space alpha over denominator cos space alpha end fraction end cell end table 

Maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell sec space alpha minus sin space alpha space tan space alpha end cell equals cell fraction numerator 1 over denominator cos space alpha end fraction minus sin space alpha times fraction numerator sin space alpha over denominator cos space alpha end fraction end cell row blank equals cell fraction numerator 1 minus sin squared alpha over denominator cos space alpha end fraction end cell end table 

Dengan mengingat identitas sin squared alpha plus cos squared alpha equals 1, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 minus sin squared alpha over denominator cos space alpha end fraction end cell equals cell fraction numerator open parentheses sin squared alpha plus cos squared alpha close parentheses minus sin squared alpha over denominator cos space alpha end fraction end cell row blank equals cell fraction numerator cos squared alpha over denominator cos space alpha end fraction end cell row blank equals cell cos space alpha end cell end table 

Jadi, bentuk paling sederhana dari sec space alpha minus sin space alpha space tan space alpha adalah cos space alpha.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Terakhir diupdate 02 Mei 2021

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Pertanyaan yang serupa

Sederhanakan tiap identitas berikut. a.

Pembahasan Soal:

Ingat!

  • sin squared x plus cos squared x equals 1
  • tan space x equals fraction numerator sin space x over denominator cos space x end fraction
  • sec space x equals fraction numerator 1 over denominator cos space x end fraction

Perhatikan perhitungan berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space A times tan space A plus cos space A end cell equals cell sin space A times fraction numerator sin space A over denominator cos space A end fraction plus cos space A end cell row blank equals cell fraction numerator sin squared space A over denominator cos space A end fraction plus cos space A end cell row blank equals cell fraction numerator sin squared space A over denominator cos space A end fraction plus fraction numerator cos squared space A over denominator cos space A end fraction end cell row blank equals cell fraction numerator sin squared space A plus cos squared space A over denominator cos space A end fraction end cell row blank equals cell fraction numerator 1 over denominator cos space A end fraction end cell row blank equals cell sec space A end cell end table

Dengan demikian bentuk sederhana dari 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 A end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank tan 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 A end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos 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 A end table adalah sec space A.

0

Roboguru

Bentuk  dapat diubah menjadi:

Pembahasan Soal:

Diketahui fraction numerator sin space x over denominator 1 plus cos space x end fraction maka dapat dijabarkan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space x over denominator 1 plus cos space x end fraction end cell equals cell fraction numerator sin space x over denominator 1 plus cos space x end fraction cross times 1 end cell row blank equals cell fraction numerator sin space x over denominator 1 plus cos space x end fraction cross times fraction numerator 1 minus cos space x over denominator 1 minus cos space x end fraction end cell row blank equals cell fraction numerator sin space x open parentheses 1 minus cos space x close parentheses over denominator 1 minus cos space x plus cos space x minus cos squared x end fraction end cell row blank equals cell fraction numerator sin space x open parentheses 1 minus cos space x close parentheses over denominator 1 minus cos squared x end fraction end cell end table

Berdasarkan identitas trigonometri yaitu sin squared x plus cos squared x equals 1 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space x over denominator 1 plus cos space x end fraction end cell equals cell fraction numerator sin space x open parentheses 1 minus cos space x close parentheses over denominator 1 minus cos squared x end fraction end cell row blank equals cell fraction numerator sin space x open parentheses 1 minus cos space x close parentheses over denominator sin squared x end fraction end cell row blank equals cell fraction numerator up diagonal strike sin space x end strike open parentheses 1 minus cos space x close parentheses over denominator up diagonal strike sin space x end strike times sin space x end fraction end cell row blank equals cell fraction numerator 1 minus cos space x over denominator sin space x end fraction end cell row blank equals cell fraction numerator 1 over denominator sin space x end fraction minus fraction numerator cos space x over denominator sin space x end fraction end cell end table

Berdasarkan definisi cotangen yaitu cot space x equals fraction numerator cos space x over denominator sin space x end fraction dan identitas trigonometri yaitu csc space x equals fraction numerator 1 over denominator sin space x end fraction maka

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space x over denominator 1 plus cos space x end fraction end cell equals cell fraction numerator 1 over denominator sin space x end fraction minus fraction numerator cos space x over denominator sin space x end fraction end cell row blank equals cell csc space x minus cot space x end cell end table

Jadi bentuk lain dari fraction numerator sin space x over denominator 1 plus cos space x end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell csc space x minus cot space x end cell end table.

Oleh karena itu, jawaban yang benar adalah B.

2

Roboguru

Buktikan bahwa  !

Pembahasan Soal:

Dengan menggunakan konsep identitas trigonometri diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin to the power of 4 x minus cos to the power of 4 x end cell equals cell left parenthesis sin squared x plus cos squared x right parenthesis left parenthesis sin squared x minus cos squared x right parenthesis end cell row blank equals cell left parenthesis sin squared x minus cos squared x right parenthesis end cell row blank equals cell sin squared x minus 1 plus sin squared x end cell row blank equals cell 2 sin squared x minus 1 end cell end table


Jadi, terbukti bahwa sin to the power of 4 x minus cos to the power of 4 x equals 2 sin squared x minus 1.

0

Roboguru

Bentuk sederhana dari  adalah ....

Pembahasan Soal:

Ingat!

begin mathsize 14px style sin squared invisible function application x plus cos squared invisible function application x equals 1 tan invisible function application x equals fraction numerator sin invisible function application x over denominator cos invisible function application x end fraction end style 

Perhatikan perhitungan berikut!

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 1 plus tan squared invisible function application x close parentheses open parentheses 1 minus cos squared invisible function application x close parentheses end cell equals cell open parentheses 1 plus fraction numerator sin squared invisible function application x over denominator cos squared invisible function application x end fraction close parentheses open parentheses sin squared invisible function application x close parentheses end cell row blank equals cell sin squared invisible function application x plus fraction numerator sin to the power of 4 invisible function application x over denominator cos squared invisible function application x end fraction end cell row blank equals cell fraction numerator sin squared invisible function application x cos squared invisible function application x plus sin to the power of 4 invisible function application x over denominator cos squared invisible function application x end fraction end cell row blank equals cell fraction numerator sin squared invisible function application x left parenthesis 1 minus sin squared invisible function application x right parenthesis plus sin to the power of 4 invisible function application x over denominator cos squared invisible function application x end fraction end cell row blank equals cell fraction numerator sin squared invisible function application x minus sin to the power of 4 invisible function application x plus sin to the power of 4 invisible function application x over denominator cos squared invisible function application x end fraction end cell row blank equals cell fraction numerator sin squared invisible function application x over denominator cos squared invisible function application x end fraction end cell row blank equals cell tan squared invisible function application x end cell end table end style 

Jadi, jawaban yang tepat adalah C.

0

Roboguru

Jawablah pertanyaan-pertanyan di bawah ini dengan benar! Buktikan Identitas-identitas trigonometri berikut!

Pembahasan Soal:

Akan dibuktikan begin mathsize 14px style fraction numerator sin x over denominator 1 plus cos x end fraction equals fraction numerator 1 minus cos x over denominator sin x end fraction end style

Ingat Bahwa begin mathsize 14px style sin squared x plus cos squared x equals 1 end style 

begin mathsize 14px style rightwards arrow fraction numerator sin x over denominator 1 plus cos x end fraction equals fraction numerator sin x over denominator 1 plus cos x end fraction fraction numerator 1 minus cos x over denominator 1 minus cos x end fraction equals fraction numerator sin x minus sin x cos x over denominator 1 minus cos squared x end fraction equals fraction numerator sin x minus sin x cos x over denominator sin squared x end fraction equals fraction numerator 1 minus cos x over denominator sin x end fraction end style

Jadi, terbukti bahwa 

begin mathsize 14px style fraction numerator sin x over denominator 1 plus cos x end fraction equals fraction numerator 1 minus cos x over denominator sin x end fraction end style.

1

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