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Pada sebuah segitiga KLM , dengan siku-siku di L .Jika sin M = 3 2 ​ dan panjang sisi KL = 10 ​ cm , tentukan panjang sisi segitiga yang lain dan nilai perbandingan trigonometri lainnya untuk sudut M !

Pada sebuah segitiga , dengan siku-siku di . Jika  dan panjang sisi , tentukan panjang sisi segitiga yang lain dan nilai perbandingan trigonometri lainnya untuk sudut !

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S. Nur

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panjang sisi , , , , , , dan .

panjang sisi KM equals 3 over 2 square root of 10 space end root space cmLM equals 5 square root of 5 space cmtan space M equals 2 over 5 square root of 5cos space M equals 1 third square root of 5, cosec space M equals 3 over 2sec space M equals 3 over 5 square root of 5, dan cotan space M equals 1 half square root of 5.

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Perhatikansegitiga berikut. Diketahui maka . Lalu, diketahuipanjang sisi maka diperoleh panjang sisi sebagai berikut. Selanjutnya, dengan menggunakan teorema pythagoras diperoleh panjang sisi sebagai berikut. Lalu,nilai perbandingan trigonometri lainnya untuk sudut sebagai berikut. Dengan demikian, panjang sisi , , , , , , dan .

Perhatikan segitiga begin mathsize 14px style KLM end style berikut.

Diketahui sin space M equals 2 over 3 maka sisi space depan space colon space sisi space miring space equals KL space colon space KM equals 2 space colon space 3. Lalu, diketahui panjang sisi begin mathsize 14px style KL equals square root of 10 space end root space cm end style maka diperoleh panjang sisi KM sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell KL space colon space KM space end cell equals cell space 2 space colon space 3 end cell row cell KL over KM end cell equals cell 2 over 3 end cell row cell fraction numerator square root of 10 over denominator KM end fraction end cell equals cell 2 over 3 end cell row cell square root of 10 cross times 3 end cell equals cell 2 KM end cell row cell 2 KM end cell equals cell 3 square root of 10 end cell row KM equals cell 3 over 2 square root of 10 space cm end cell end table end style

Selanjutnya, dengan menggunakan teorema pythagoras diperoleh panjang sisi LM sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell KM squared end cell equals cell KL squared plus LM squared end cell row cell open parentheses 3 over 2 square root of 10 close parentheses squared end cell equals cell open parentheses square root of 10 close parentheses squared plus LM squared end cell row cell 9 over 4 cross times 10 end cell equals cell 10 plus LM squared end cell row cell LM squared end cell equals cell 90 over 4 minus 10 end cell row blank equals cell 90 over 4 minus 40 over 4 end cell row blank equals cell 50 over 4 end cell row LM equals cell square root of 50 over 4 end root end cell row blank equals cell fraction numerator 5 square root of 2 over denominator 2 end fraction cm end cell end table

Lalu, nilai perbandingan trigonometri lainnya untuk sudut M sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space M end cell equals cell fraction numerator sisi space depan over denominator sisi space samping end fraction end cell row blank equals cell KL over LM end cell row blank equals cell fraction numerator square root of 10 over denominator begin display style 5 over 2 end style square root of 2 end fraction end cell row blank equals cell fraction numerator square root of 10 over denominator begin display style 5 over 2 end style square root of 2 end fraction times fraction numerator square root of 2 over denominator square root of 2 end fraction end cell row blank equals cell fraction numerator square root of 20 over denominator 5 end fraction end cell row blank equals cell fraction numerator 2 square root of 5 over denominator 5 end fraction end cell row blank equals cell 2 over 5 square root of 5 end cell end table
 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space M end cell equals cell fraction numerator sisi space samping over denominator sisi space miring end fraction end cell row blank equals cell LM over KM end cell row blank equals cell fraction numerator begin display style 5 over 2 end style square root of 5 over denominator begin display style 3 over 2 end style 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 end cell row blank equals cell fraction numerator 5 square root of 2 over denominator 3 square root of 10 end fraction 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 cross times 2 square root of 5 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 space end cell end table
 

table attributes columnalign right center left columnspacing 0px end attributes row cell cosec space M end cell equals cell fraction numerator 1 over denominator sin space M 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
 

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

table attributes columnalign right center left columnspacing 0px end attributes row cell cotan space M end cell equals cell fraction numerator 1 over denominator tan space M end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style 2 over 5 end style square root of 5 end fraction end cell row blank equals cell fraction numerator 5 over denominator 2 square root of 5 end fraction times fraction numerator square root of 5 over denominator square root of 5 end fraction end cell row blank equals cell fraction numerator 5 square root of 5 over denominator 10 end fraction end cell row blank equals cell 1 half square root of 5 end cell end table


Dengan demikian, panjang sisi KM equals 3 over 2 square root of 10 space end root space cmLM equals 5 square root of 5 space cmtan space M equals 2 over 5 square root of 5cos space M equals 1 third square root of 5, cosec space M equals 3 over 2sec space M equals 3 over 5 square root of 5, dan cotan space M equals 1 half square root of 5.

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