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Perhatikan gambar berikut!   Dengan aturan cosinus yaitu:   tunjukkan bahwa !

Pertanyaan

Perhatikan gambar berikut!



 

Dengan aturan cosinus yaitu:

open vertical bar PQ with rightwards arrow on top close vertical bar squared equals open vertical bar OP with rightwards arrow on top close vertical bar squared plus open vertical bar OQ with rightwards arrow on top close vertical bar squared minus 2 open vertical bar OP with rightwards arrow on top close vertical bar open vertical bar OQ with rightwards arrow on top close vertical bar cos space theta 

tunjukkan bahwa a with rightwards arrow on top times b with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar space cos space theta!space 

Pembahasan Soal:

Jika terdapat titik straight A open parentheses a subscript 1 comma space a subscript 2 close parentheses dan titik straight B open parentheses b subscript 1 comma space b subscript 2 close parentheses, maka berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell open parentheses a subscript 1 comma space a subscript 2 close parentheses comma space b with rightwards arrow on top equals open parentheses b subscript 1 comma space b subscript 2 close parentheses end cell row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell open parentheses a subscript 1 comma space a subscript 2 close parentheses times open parentheses b subscript 1 comma space b subscript 2 close parentheses end cell row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 end cell row cell AB with rightwards arrow on top end cell equals cell b with rightwards arrow on top minus a with rightwards arrow on top end cell row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell square root of a subscript 1 squared plus a subscript 2 squared end root end cell end table  

Keterangan:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell perkalian space titik space a with rightwards arrow on top space dan space b with rightwards arrow on top end cell row cell AB with rightwards arrow on top end cell equals cell vektor space posisi space dari space titik space straight B space ke space titik space straight A end cell row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell panjang space a with rightwards arrow on top end cell end table 

Berdasarkan gambar, diketahui:

straight O open parentheses 0 comma space 0 close parentheses comma space straight P open parentheses a subscript 1 comma space a subscript 2 close parentheses comma space straight Q open parentheses b subscript 1 comma space b subscript 2 close parentheses space stack O P with rightwards arrow on top equals a with rightwards arrow on top stack O Q with rightwards arrow on top equals b with rightwards arrow on top  

Bukti:

begin mathsize 11px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar PQ with rightwards arrow on top close vertical bar squared end cell equals cell open vertical bar OP with rightwards arrow on top close vertical bar squared plus open vertical bar OQ with rightwards arrow on top close vertical bar squared minus 2 open vertical bar OP with rightwards arrow on top close vertical bar open vertical bar OQ with rightwards arrow on top close vertical bar cos space theta end cell row cell open parentheses square root of open parentheses b subscript 1 minus a subscript 1 close parentheses squared plus open parentheses b subscript 2 minus a subscript 2 close parentheses squared end root close parentheses squared end cell equals cell open parentheses square root of a subscript 1 squared plus a subscript 2 squared end root close parentheses squared plus open parentheses square root of b subscript 1 squared plus b subscript 2 squared end root close parentheses squared minus 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell open parentheses b subscript 1 minus a subscript 1 close parentheses squared plus open parentheses b subscript 2 minus a subscript 2 close parentheses squared end cell equals cell a subscript 1 squared plus a subscript 2 squared plus b subscript 1 squared plus b subscript 2 squared minus 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell b subscript 1 squared minus 2 a subscript 1 b subscript 1 plus a subscript 1 squared plus b subscript 2 squared minus 2 a subscript 2 b subscript 2 plus a subscript 2 squared end cell equals cell a subscript 1 squared plus a subscript 2 squared plus b subscript 1 squared plus b subscript 2 squared minus 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell negative 2 a subscript 1 b subscript 1 minus 2 a subscript 2 b subscript 2 end cell equals cell negative 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell negative 2 open parentheses a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 close parentheses end cell equals cell negative 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 end cell equals cell square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar cos space theta end cell end table end style 

Jadi, terbukti bahwa a with rightwards arrow on top times b with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar space cos space theta.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 03 Juni 2021

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