Suatu segitiga ABC siku-siku di B, jika sin C=32 dan Panjang sisi AB=10 cm, tentukan Panjang sisi segitiga yang lain dan nilai perbandingan trigonometri lainnya.

Pertanyaan

Suatu segitiga ABC siku-siku di B, jika $\sin\;\mathrm C=\frac23$ dan Panjang sisi $\mathrm{AB}=\sqrt{10}\;\mathrm{cm}$ , tentukan Panjang sisi segitiga yang lain dan nilai perbandingan trigonometri lainnya.

R. Risda

Master Teacher

Jawaban terverifikasi

Jawaban

panjang

BC equals 5 over 2 square root of 2 space cm

AC equals 3 over 2 square root of 10 space cm

cos space C equals 1 third square root of 5

tan space C equals 2 over 5 square root of 5

sec space straight C equals 3 over 5 square root of 5

dan $\mathrm{ctg}\;\mathrm C=\frac12\sqrt5$ .

Pembahasan

Perbandingan nilai trigonometri pada segitiga siku-siku.

segitiga ABC siku-siku di B maka AC sisi miring.

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight C end cell equals cell fraction numerator sisi space depan space straight C over denominator sisi space miring end fraction end cell row cell 2 over 3 end cell equals cell AB over AC end cell row cell 2 over 3 end cell equals cell fraction numerator square root of 10 space cm over denominator AC end fraction end cell row AC equals cell fraction numerator 3 square root of 10 over denominator 2 end fraction space cm end cell row blank blank blank end table$

Dengan teorema pythagoras:

$\begin{array}{rcl}\mathrm{BC}&=&\sqrt{\mathrm{AC}^2-\mathrm{AB}^2}\\&=&\sqrt{{(\frac32\sqrt{10})}^2-{(\sqrt{10})}^2}\\&=&\sqrt{\frac94\cdot10-10}\\&=&\sqrt{\frac{90}4-\frac{40}4}\\&=&\sqrt{\frac{50}4}\\&=&\frac12\sqrt{50}\\&=&\frac12\sqrt{2\cdot25}\\&=&\frac52\sqrt2\;\mathrm{cm}\end{array}$

Perbandingan trigonometeri lainnya:

Cos C yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space straight C end cell equals cell fraction numerator sisi space samping space straight C over denominator sisi space miring end fraction end cell row blank equals cell BC over AC end cell row blank equals cell fraction numerator begin display style 5 over 2 square root of 2 end style over denominator begin display style 3 over 2 square root of 10 end style end fraction end cell row blank equals cell fraction numerator 5 square root of 2 over denominator 3 square root of 10 end fraction end cell row blank equals cell fraction numerator 5 square root of 2 over denominator 3 square root of 10 end fraction cross times fraction numerator square root of 10 over denominator square root of 10 end fraction end cell row blank equals cell fraction numerator 5 square root of 20 over denominator 30 end fraction end cell row blank equals cell fraction numerator 5 square root of 4 cross times 5 end root over denominator 30 end fraction end cell row blank equals cell fraction numerator 10 square root of 5 over denominator 30 end fraction end cell row blank equals cell 1 third square root of 5 end cell end table$

tan C yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan space straight C end cell equals cell fraction numerator sin space straight C over denominator cos space straight C end fraction end cell row blank equals cell fraction numerator begin display style 2 over 3 end style over denominator begin display style 1 third square root of 5 end style end fraction end cell row blank equals cell fraction numerator 2 over denominator square root of 5 end fraction end cell row blank equals cell 2 over 5 square root of 5 end cell end table$

Cosec C yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell cosec space straight C end cell equals cell fraction numerator 1 over denominator sin space straight C end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style 2 over 3 end style end fraction end cell row blank equals cell 3 over 2 end cell end table$

Secan C yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sec space straight C end cell equals cell fraction numerator 1 over denominator cos space straight C end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style fraction numerator square root of 5 over denominator 3 end fraction end style end fraction end cell row blank equals cell fraction numerator 3 over denominator square root of 5 end fraction end cell row blank equals cell fraction numerator 3 cross times square root of 5 over denominator square root of 5 cross times square root of 5 end fraction end cell row blank equals cell 3 over 5 square root of 5 end cell end table$

Cotangen C yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell cot space straight C end cell equals cell fraction numerator 1 over denominator tan space straight C end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style fraction numerator 2 over denominator square root of 5 end fraction end style end fraction end cell row blank equals cell fraction numerator square root of 5 over denominator 2 end fraction end cell row blank equals cell 1 half square root of 5 end cell end table$

Dengan demikian, panjang BC equals 5 over 2 square root of 2 space cm , AC equals 3 over 2 square root of 10 space cm , cos space C equals 1 third square root of 5 , tan space C equals 2 over 5 square root of 5 , csc space straight C equals 3 over 2 , sec space straight C equals 3 over 5 square root of 5 dan $\mathrm{ctg}\;\mathrm C=\frac12\sqrt5$ .

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