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Buktikan identitas trigonometri berikut ini. a. 3 ( sin 4 θ + cos 4 θ ) − 2 ( sin 6 θ + cos 6 θ ) = 1

Buktikan identitas trigonometri berikut ini.

a. 

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Pembahasan

Ingat! Akan dibuktikan Perhatikan Selanjutnya perhatikan juga perhitungan berikut: Dengan menggunakan persamaan dan diperoleh Dengan demikian terbukti bahwa: .

Ingat!

sin squared theta plus cos squared theta equals 1

Akan dibuktikan

3 open parentheses sin to the power of 4 theta plus cos to the power of 4 theta close parentheses minus 2 open parentheses sin to the power of 6 theta plus cos to the power of 6 theta close parentheses equals 1

Perhatikan

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin to the power of 4 theta plus cos to the power of 4 theta end cell row blank equals cell open parentheses sin to the power of 4 theta plus cos to the power of 4 theta close parentheses times 1 end cell row blank equals cell open parentheses sin to the power of 4 theta plus cos to the power of 4 theta close parentheses times open parentheses sin squared theta plus cos squared theta close parentheses end cell row blank equals cell sin to the power of 6 theta plus sin to the power of 4 theta space cos squared theta plus sin squared theta space cos to the power of 4 theta plus cos to the power of 6 theta end cell row blank equals cell sin to the power of 6 theta plus cos to the power of 6 theta plus sin to the power of 4 theta space cos squared theta plus sin squared theta space cos to the power of 4 theta end cell row blank equals cell sin to the power of 6 theta plus cos to the power of 6 theta plus sin squared theta space cos squared theta open parentheses sin squared theta plus space cos squared theta close parentheses end cell row blank equals cell sin to the power of 6 theta plus cos to the power of 6 theta plus sin squared theta space cos squared theta times 1 end cell row blank equals cell sin to the power of 6 theta plus cos to the power of 6 theta plus sin squared theta space cos squared theta space.... left parenthesis space 1 right parenthesis end cell end table

Selanjutnya perhatikan juga perhitungan berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin squared theta plus cos squared theta end cell equals 1 row cell open parentheses sin squared theta plus cos squared theta close parentheses cubed end cell equals cell 1 cubed end cell row cell sin to the power of 6 theta plus 3 sin to the power of 4 theta cos squared theta plus 3 sin squared theta cos to the power of 4 theta plus cos to the power of 6 end cell equals 1 row cell sin to the power of 6 theta plus cos to the power of 6 plus 3 sin to the power of 4 theta cos squared theta plus 3 sin squared theta cos to the power of 4 theta end cell equals 1 row cell sin to the power of 6 theta plus cos to the power of 6 plus 3 sin squared theta cos squared theta open parentheses sin squared theta plus cos squared theta close parentheses end cell equals 1 row cell sin to the power of 6 theta plus cos to the power of 6 plus 3 sin squared theta cos squared theta times 1 end cell equals 1 row cell sin to the power of 6 theta plus cos to the power of 6 plus 3 sin squared theta cos squared theta end cell equals cell 1 space space.... open parentheses space 2 space close parentheses end cell end table

Dengan menggunakan persamaan open parentheses 1 close parentheses dan open parentheses 2 close parentheses diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 open parentheses sin to the power of 4 theta plus cos to the power of 4 theta close parentheses minus 2 open parentheses sin to the power of 6 theta plus cos to the power of 6 theta close parentheses end cell row blank equals cell 3 open parentheses sin to the power of 6 theta plus cos to the power of 6 theta plus sin squared theta space cos squared theta close parentheses minus 2 open parentheses sin to the power of 6 theta plus cos to the power of 6 theta close parentheses end cell row blank equals cell 3 sin to the power of 6 theta plus 3 cos to the power of 6 theta plus 3 sin squared theta space cos squared theta minus 2 sin to the power of 6 theta minus 2 cos to the power of 6 theta end cell row blank equals cell 3 sin to the power of 6 theta minus 2 sin to the power of 6 theta plus 3 cos to the power of 6 theta minus 2 cos to the power of 6 theta plus 3 sin squared theta space cos squared theta end cell row blank equals cell sin to the power of 6 theta plus cos to the power of 6 theta plus 3 sin squared theta space cos squared theta end cell row blank equals 1 end table

Dengan demikian terbukti bahwa:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses sin to the power of 4 theta plus cos to the power of 4 theta close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses sin to the power of 6 theta plus cos to the power of 6 theta close parentheses end cell 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 1 end table.

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