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Buktikan persamaan berikut ini! 5 cos2 A=5−5 sin2 A

Pertanyaan

Buktikan persamaan berikut ini!

begin mathsize 14px style 5 space cos squared space straight A equals 5 minus 5 space sin squared space straight A end style  

Z. Apriani

Master Teacher

Mahasiswa/Alumni Institut Teknologi Bandung

Jawaban terverifikasi

Jawaban

terbukti bahwa  begin mathsize 14px style 5 space cos squared space straight A equals 5 minus 5 space sin squared space straight A end style.

Pembahasan

Perhatikan gambar berikut!

Segitiga ABC adalah segitiga siku-siku, sehingga berlaku teorema Pythagoras:

begin mathsize 14px style r squared equals x squared plus y squared x squared equals r squared minus y squared y squared equals r squared minus x squared end style 

Pada segitiga ABC berlaku:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space alpha end cell equals cell depan over miring end cell row blank equals cell y over r end cell row cell cos space alpha end cell equals cell samping over miring end cell row blank equals cell x over r end cell end table end style 

Sehingga,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 5 space cos squared space straight A end cell equals cell 5 minus 5 space sin squared space straight A end cell row blank equals cell 5 minus 5 times open parentheses y over r close parentheses squared end cell row blank equals cell 5 minus fraction numerator 5 y squared over denominator r squared end fraction end cell row blank equals cell fraction numerator 5 r squared minus 5 y squared over denominator r squared end fraction end cell row blank equals cell fraction numerator 5 open parentheses r squared minus y squared close parentheses over denominator r squared end fraction end cell row blank equals cell fraction numerator 5 x squared over denominator r squared end fraction end cell row blank equals cell 5 space cos squared space straight A space end cell row blank blank blank end table end style 

Jadi, terbukti bahwa  begin mathsize 14px style 5 space cos squared space straight A equals 5 minus 5 space sin squared space straight A end style.

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