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Diketahui , panjang sisi , , . Tentukan panjang sisi BC dan sisi AB!

Pertanyaan

Diketahui triangle ABC, panjang sisi AC equals 8 space cmangle straight A equals 60 degreeangle straight B equals 75 degree. Tentukan panjang sisi BC dan sisi AB!

Pembahasan Soal:

Perhatikan gambar triangle ABC berikut!

Nilai sinus sudut B adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 75 degree end cell equals cell sin space open parentheses 45 degree plus 30 degree close parentheses end cell row blank equals cell sin space 45 degree times cos space 30 degree plus cos space 45 degree times sin space 30 degree end cell row blank equals cell 1 half square root of 2 times 1 half square root of 3 plus 1 half square root of 2 times 1 half end cell row blank equals cell 1 fourth square root of 2 open parentheses square root of 3 plus 1 close parentheses end cell end table

Dengan aturan sinus diperoleh panjang sisi BC, yaitu

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator AC over denominator sin space straight B end fraction end cell equals cell fraction numerator BC over denominator sin space straight A end fraction end cell row cell fraction numerator 8 over denominator sin space 75 degree end fraction end cell equals cell fraction numerator BC over denominator sin space 60 degree end fraction end cell row cell fraction numerator 8 over denominator begin display style 1 fourth end style square root of 2 open parentheses square root of 3 plus 1 close parentheses end fraction end cell equals cell fraction numerator BC over denominator 1 half square root of 3 end fraction end cell row cell 1 fourth square root of 2 open parentheses square root of 3 plus 1 close parentheses times BC end cell equals cell 8 times 1 half square root of 3 end cell row cell 1 fourth square root of 2 open parentheses square root of 3 plus 1 close parentheses times BC end cell equals cell 4 square root of 3 end cell row cell square root of 2 open parentheses square root of 3 plus 1 close parentheses times BC end cell equals cell 16 square root of 3 end cell row BC equals cell fraction numerator 16 square root of 3 over denominator square root of 2 open parentheses square root of 3 plus 1 close parentheses end fraction end cell row BC equals cell fraction numerator 16 square root of 3 over denominator square root of 2 open parentheses square root of 3 plus 1 close parentheses end fraction times fraction numerator square root of 2 open parentheses square root of 3 minus 1 close parentheses over denominator square root of 2 open parentheses square root of 3 minus 1 close parentheses end fraction end cell row BC equals cell fraction numerator 16 square root of 6 open parentheses square root of 3 minus 1 close parentheses over denominator 2 open parentheses 3 minus 1 close parentheses end fraction end cell row BC equals cell fraction numerator 16 open parentheses square root of 18 minus square root of 6 close parentheses over denominator 4 end fraction end cell row BC equals cell 4 open parentheses 3 square root of 2 minus square root of 6 close parentheses space cm end cell end table

Jumlah sudut dalam segitiga adalah 180 degree, sehingga angle straight C equals 45 degree. Dengan aturan sinus diperoleh panjang sisi AB sebagai berikut,

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator AB over denominator sin space straight C end fraction end cell equals cell fraction numerator AC over denominator sin space straight B end fraction end cell row cell fraction numerator AB over denominator sin space 45 degree end fraction end cell equals cell fraction numerator 8 over denominator sin space 75 degree end fraction end cell row cell fraction numerator AB over denominator begin display style 1 half end style square root of 2 end fraction end cell equals cell fraction numerator 8 over denominator begin display style 1 fourth end style square root of 2 open parentheses square root of 3 plus 1 close parentheses end fraction end cell row cell 1 fourth square root of 2 open parentheses square root of 3 plus 1 close parentheses times AB end cell equals cell 1 half square root of 2 times 8 end cell row cell 1 fourth square root of 2 open parentheses square root of 3 plus 1 close parentheses times AB end cell equals cell 4 square root of 2 end cell row AB equals cell fraction numerator 4 square root of 2 over denominator 1 fourth square root of 2 open parentheses square root of 3 plus 1 close parentheses end fraction end cell row AB equals cell fraction numerator 16 over denominator square root of 3 plus 1 end fraction times fraction numerator square root of 3 minus 1 over denominator square root of 3 minus 1 end fraction end cell row AB equals cell fraction numerator 16 open parentheses square root of 3 minus 1 close parentheses over denominator 3 minus 1 end fraction end cell row AB equals cell 8 open parentheses square root of 3 minus 1 close parentheses space cm end cell end table 

Jadi, BC equals 4 open parentheses 3 square root of 2 minus square root of 6 close parentheses space cm dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank AB 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 8 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of 3 minus 1 close parentheses end cell 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 cm end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Sri

Mahasiswa/Alumni Universitas Muhammadiyah Purworejo

Terakhir diupdate 04 Juni 2021

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Pembahasan Soal:

Perhatikan penghitungan berikut!

begin mathsize 14px style Jarak space tempuh space selama space 4 space jam space berlayar equals 45 space mil divided by jam cross times 4 space jam equals 180 space mil  misalkan colon x equals jarak space kapal space terhadap space arah space timur space pelabuhan sin space 60 degree equals fraction numerator x over denominator 180 space mil end fraction x equals sin space 60 degree times 180 space mil x equals 1 half square root of 13 times 180 space mil x equals 90 square root of 3 mil end style 

Oleh karena itu, jawaban yang benar adalah C. undefined 

0

Roboguru

Diketahui  dengan , , dan . Tentukan: a. Panjang sisi AB b.

Pembahasan Soal:

a. Panjang sisi AB

Dengan menggunakan aturan cosinus, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell AB squared end cell equals cell AC squared plus BC squared minus 2 times AC times BC times cos space angle ACB end cell row cell AB squared end cell equals cell 12 squared plus 8 squared minus 2 times 12 times 8 space cos space 120 degree end cell row cell AB squared end cell equals cell 144 plus 64 minus 192 times open parentheses negative 1 half close parentheses end cell row cell AB squared end cell equals cell 208 minus open parentheses negative 96 close parentheses end cell row cell AB squared end cell equals cell 208 plus 96 end cell row cell AB squared end cell equals 304 row AB equals cell plus-or-minus square root of 304 end cell row AB equals cell plus-or-minus 4 square root of 19 end cell row AB equals cell 4 square root of 19 space cm end cell end table

Maka, panjang sisi AB adalah 4 square root of 19 space cm.

 

b. sin space angle ABC

Dengan menggunakan aturan sinus, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space angle ABC over denominator AC end fraction end cell equals cell fraction numerator sin space angle ACB over denominator AB end fraction end cell row cell fraction numerator sin space angle ABC over denominator 12 space cm end fraction end cell equals cell fraction numerator sin space 120 degree over denominator 4 square root of 19 space cm end fraction end cell row cell sin space angle ABC end cell equals cell fraction numerator 12 space cm times sin space 120 degree over denominator 4 square root of 19 space cm end fraction end cell row cell sin space angle ABC end cell equals cell fraction numerator 12 space cm times begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 4 square root of 19 space cm end fraction end cell row cell sin space angle ABC end cell equals cell fraction numerator 6 square root of 3 space cm over denominator 4 square root of 19 space cm end fraction end cell row cell sin space angle ABC end cell equals cell fraction numerator 6 square root of 3 over denominator 4 square root of 19 end fraction times fraction numerator square root of 19 over denominator square root of 19 end fraction end cell row cell sin space angle ABC end cell equals cell fraction numerator 6 square root of 57 over denominator 4 times 19 end fraction end cell row cell sin space angle ABC end cell equals cell fraction numerator 6 square root of 57 over denominator 76 end fraction end cell row cell sin space angle ABC end cell equals cell fraction numerator 3 square root of 57 over denominator 38 end fraction end cell row cell sin space angle ABC end cell equals cell arc space sin space open parentheses fraction numerator 3 square root of 57 over denominator 38 end fraction close parentheses end cell row cell sin space angle ABC end cell equals cell 36 comma 59 degree end cell end table

Maka, sin space angle ABC adalah 36 comma 59 degree.

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Roboguru

Pada gambar di bawah ini, PQR adalah segitiga siku-siku di Q dan sama kaki. Jika diberikan  dan ,   a. tunjukkanlah bahwa

Pembahasan Soal:

0

Roboguru

Jika diketahui segitiga ABC, dengan ukuran panjang sisi , besar  dan besar . Hitung panjang sisi  dan !

Pembahasan Soal:

Mencari sudut begin mathsize 14px style A end style :

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell angle A end cell equals cell 180 to the power of ring operator minus angle B minus angle C end cell row blank equals cell 180 to the power of ring operator minus 45 to the power of ring operator minus 105 to the power of ring operator end cell row blank equals cell 30 to the power of ring operator end cell end table end style

Dengan aturan sinus, panjang sisi begin mathsize 14px style a end style :

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator a over denominator sin space A end fraction end cell equals cell fraction numerator b over denominator sin space B end fraction end cell row cell fraction numerator a over denominator sin space 30 to the power of ring operator end fraction end cell equals cell fraction numerator 20 over denominator sin space 45 to the power of ring operator end fraction end cell row a equals cell fraction numerator 20 over denominator begin display style 1 half end style square root of 2 end fraction cross times 1 half end cell row a equals cell 10 square root of 2 end cell end table end style

Dengan aturan cosinus, panjang sisi begin mathsize 14px style c end style :

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a squared end cell equals cell b squared plus c squared minus 2 b c times cos space A end cell row cell left parenthesis 10 square root of 2 right parenthesis squared end cell equals cell 20 to the power of 2 end exponent plus c squared minus 2 times left parenthesis 20 right parenthesis times c times 1 half square root of 3 end cell row 200 equals cell 400 plus c squared minus 20 square root of 3 c end cell row 0 equals cell c squared minus 20 square root of 3 c plus 200 end cell end table end style

Dengan rumus kuadratik :

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell c subscript 1 comma 2 end subscript end cell equals cell fraction numerator negative left parenthesis negative 20 square root of 3 right parenthesis plus-or-minus square root of open parentheses 20 square root of 3 close parentheses squared minus 4 times 1 times 200 end root over denominator 2 times 1 end fraction end cell row blank equals cell fraction numerator 20 square root of 3 plus-or-minus 20 over denominator 2 end fraction end cell row blank equals cell 10 square root of 3 plus-or-minus 10 end cell end table end style

Karena besar begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank angle end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table table attributes columnalign right center left columnspacing 0px end attributes row blank greater than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank angle end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank B end table table attributes columnalign right center left columnspacing 0px end attributes row blank greater than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank angle end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank A end table end style, maka panjang sisi undefined

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Roboguru

Perhatikan gambar segiempat ABCD berikut! Panjang BC = ....

Pembahasan Soal:

Perhatikan space segitiga space ADB comma space cari space panjang space BD space melalui space aturan space sinus  fraction numerator BD over denominator sin space straight A end fraction equals fraction numerator AD over denominator sin space straight B end fraction  fraction numerator BD over denominator sin space 30 to the power of straight o end fraction equals fraction numerator 10 over denominator sin space 45 to the power of straight o end fraction  fraction numerator BD over denominator begin display style 1 half end style end fraction equals fraction numerator 10 over denominator begin display style 1 half square root of 2 end style end fraction  BD equals fraction numerator 10 over denominator square root of 2 end fraction  BD equals 5 square root of 2 space space cm    Kemudian space perhatikan space segitiga space BCD space dan space melalui space aturan space cosinus space cari space BC comma  BC squared equals BD squared plus CD squared minus 2 BC. BDcos invisible function application straight D  BC squared equals left parenthesis 5 square root of 2 right parenthesis squared plus left parenthesis 10 square root of 2 right parenthesis squared minus 2 left parenthesis 5 square root of 2 right parenthesis left parenthesis 10 square root of 2 right parenthesis cos invisible function application 60 degree  BC squared equals 50 plus 200 minus 200 left parenthesis 1 half right parenthesis  BC squared equals 250 minus 100  BC squared equals 150  BC equals square root of 150  BC equals 5 square root of 6 space space cm

1

Roboguru

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