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Tunjukkan bahwa persamaan sin x1−cos x​=1+cos xsin x​ dipenuhi pada semua x kecuali kelipatan π.

Pertanyaan

Tunjukkan bahwa persamaan fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction dipenuhi pada semua x kecuali kelipatan straight pi.

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

berdasarkan hitungan di atas terbukti bahwa persamaan fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction dipenuhi pada semua x kecuali kelipatan straight pi.

Pembahasan

Diketahui fraction numerator 1 minus cos space x over denominator sin space x end fraction maka diperoleh sebagai berikut.

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

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

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

Jadi berdasarkan hitungan di atas terbukti bahwa fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction agar persamaan fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction terpenuhi maka sin space x not equal to 0 sehingga

begin mathsize 12px style table row blank blank cell sin space x not equal to 0 end cell blank row blank cell sin space x not equal to sin space 0 degree end cell blank cell sin space x not equal to sin open parentheses 180 degree minus 0 degree close parentheses end cell row blank cell x not equal to 0 degree plus-or-minus k times 360 degree end cell blank cell x not equal to 180 degree plus-or-minus k times 360 degree end cell row cell k equals 0 space rightwards arrow end cell cell x not equal to 0 degree plus-or-minus 0 times 360 degree end cell blank cell x not equal to 180 plus-or-minus 0 times 360 end cell row blank cell x not equal to 0 plus-or-minus 0 end cell blank cell x not equal to 180 plus-or-minus 0 end cell row blank cell x not equal to 0 degree end cell blank cell x not equal to 180 degree end cell row cell k equals 1 space rightwards arrow end cell cell x not equal to 0 plus-or-minus 1 times 360 end cell blank blank row blank cell x not equal to 0 plus-or-minus 360 end cell blank blank row blank cell x not equal to 360 degree end cell blank blank end table end style

Jadi berdasarkan hitungan di atas terbukti bahwa persamaan fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction dipenuhi pada semua x kecuali kelipatan straight pi.

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