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Balok  dengan , , dan  serta  titik tengah . Tentu...

Balok begin mathsize 14px style text ABCD.EFGH end text end style dengan begin mathsize 14px style text AB = 4 cm end text end style, begin mathsize 14px style text BC = 10 cm end text end style, dan begin mathsize 14px style text CG = 6 cm end text end style serta begin mathsize 14px style text P end text end style titik tengah begin mathsize 14px style text AB end text end style. Tentukan nilai sinus sudut antara begin mathsize 14px style text EP end text end style dan begin mathsize 14px style text PG end text end style!

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Jawaban:

Perhatikan perhitungan berikut ini.

Ingat, aturan cosinus, teorema Pythagoras dan fungsi identitas trigonometri:

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 times b times c times text Cos A end text end cell row cell c squared end cell equals cell a squared plus b squared end cell row 1 equals cell sin squared alpha plus cos squared alpha end cell end table end style

Diketahui sebuah balok begin mathsize 14px style text ABCD.EFGH end text end style dengan begin mathsize 14px style text AB = 4 cm end text end style, begin mathsize 14px style text BC = 10 cm end text end style, dan begin mathsize 14px style text CG = 6 cm end text end style serta begin mathsize 14px style text P end text end style titik tengah begin mathsize 14px style text AB end text end style maka balok tersebut dapat dibentuk sebagai berikut ini:

Untuk mencari sudut sinus yang terbentuk oleh garis begin mathsize 14px style text EP end text end style dan begin mathsize 14px style text PG end text end style terlebih dahulu dicari nilasi cosinus sudutnya, dicari dengan menggunakan aturan cosinus, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell text EG end text squared end cell equals cell text EP end text squared plus text PG end text squared minus 2 times text EP end text times text PG end text times cos text  P end text end cell row cell 2 times text EP end text times text PG end text times cos text  P end text end cell equals cell text EP end text squared plus text PG end text squared minus text EG end text squared end cell row cell cos text  P end text end cell equals cell fraction numerator text EP end text squared plus text PG end text squared minus text EG end text squared over denominator 2 times text EP end text times text PG end text end fraction end cell end table end style

Panjang begin mathsize 14px style text PG end text end style:

Perhatikan, dengan menggunakan teorema Pythagoras maka:

begin mathsize 14px style text PG end text squared equals text PC end text squared plus text CG end text squared end style

terlebih dahulu dicari panjang begin mathsize 14px style text PC end text end style, yaitu:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell text PC end text squared end cell equals cell text PB end text squared plus text BC end text squared end cell row cell text PC end text squared end cell equals cell 2 squared plus 10 squared end cell row cell text PC end text squared end cell equals cell 4 plus 100 end cell row cell text PC end text squared end cell equals 104 row cell text PC end text end cell equals cell plus-or-minus square root of 104 end cell end table end style

Didapatkan panjang begin mathsize 14px style text PC end text end style adalah begin mathsize 14px style plus-or-minus square root of 104 end style, karena suatu panjang tidak mungkin negatif maka dipilih begin mathsize 14px style square root of 104 end style. Sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell text PG end text squared end cell equals cell text PC end text squared plus text CG end text squared end cell row cell text PG end text squared end cell equals cell open parentheses square root of 104 close parentheses squared plus 6 squared end cell row cell text PG end text squared end cell equals cell 104 plus 36 end cell row cell text PG end text squared end cell equals 140 row cell text PG end text end cell equals cell plus-or-minus square root of 140 end cell end table end style

Didapatkan panjang begin mathsize 14px style text PG end text end style adalah begin mathsize 14px style plus-or-minus square root of 140 end style, karena suatu panjang tidak mungkin negatif maka dipilih begin mathsize 14px style 140 end style.

Panjang begin mathsize 14px style text EP end text end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell text EP end text squared end cell equals cell text EA end text squared plus text AP end text squared end cell row cell text EP end text squared end cell equals cell 6 squared plus 2 squared end cell row cell text EP end text squared end cell equals cell 36 plus 4 end cell row cell text EP end text squared end cell equals 40 row cell text EP end text end cell equals cell plus-or-minus square root of 40 end cell end table end style

Didapatkan panjang begin mathsize 14px style text EP end text end style adalah begin mathsize 14px style plus-or-minus square root of 40 end style, karena suatu panjang tidak mungkin negatif maka dipilih begin mathsize 14px style square root of 40 end style.

Panjang undefined:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell text EG end text squared end cell equals cell text EH end text squared plus text GH end text squared end cell row cell text EG end text squared end cell equals cell 10 squared plus 4 squared end cell row cell text EG end text squared end cell equals cell 100 plus 16 end cell row cell text EG end text squared end cell equals 116 row cell text EG end text end cell equals cell plus-or-minus square root of 116 end cell end table end style

Didapatkan panjang undefined adalah begin mathsize 14px style plus-or-minus square root of 116 end style, karena suatu panjang tidak mungkin negatif maka dipilih begin mathsize 14px style square root of 116 end style.

Sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos text  P end text end cell equals cell fraction numerator text EP end text squared plus text PG end text squared minus text EG end text squared over denominator 2 times text EP end text times text PG end text end fraction end cell row blank equals cell fraction numerator open parentheses square root of 40 close parentheses squared plus open parentheses square root of 140 close parentheses squared minus open parentheses square root of 116 close parentheses squared over denominator 2 open parentheses square root of 40 close parentheses open parentheses square root of 140 close parentheses end fraction end cell row blank equals cell fraction numerator 40 plus 140 minus 116 over denominator 2 square root of 40 times 140 end root end fraction end cell row blank equals cell fraction numerator 64 over denominator 2 square root of 5600 end fraction end cell row blank equals cell fraction numerator 32 over denominator square root of 5600 end fraction end cell end table end style

Didapatkan nilai begin mathsize 14px style cos space text P end text end style adalah begin mathsize 14px style fraction numerator 32 over denominator square root of 5600 end fraction end style, maka nilai begin mathsize 14px style text sin P end text end style yaitu:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin squared space text P end text space plus space cos squared space text P end text space end cell equals cell space 1 end cell row cell sin squared space text P end text end cell equals cell 1 minus cos squared space text P end text end cell row cell sin space text P end text end cell equals cell square root of 1 minus cos squared space text P end text end root end cell row cell sin space text P end text end cell equals cell square root of 1 minus open parentheses fraction numerator 32 over denominator square root of 5600 end fraction close parentheses squared end root end cell row cell sin space text P end text end cell equals cell square root of 1 minus open parentheses 32 squared over open parentheses square root of 5600 close parentheses squared close parentheses end root end cell row cell sin space text P end text end cell equals cell square root of 1 minus open parentheses 1024 over 5600 close parentheses end root end cell row cell sin space text P end text end cell equals cell square root of fraction numerator 5600 minus 1024 over denominator 5600 end fraction end root end cell row cell sin space text P end text end cell equals cell square root of 4576 over 5600 end root end cell row cell sin space text P end text end cell equals cell 0 comma 904 end cell end table end style

Jadi, nilai sinus sudut antara begin mathsize 14px style text EP end text end style dan begin mathsize 14px style text PG end text end style adalah begin mathsize 14px style 0 comma 904 end style.

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