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Tanpa menggunakan tabel trigonometri, buktikan: sin 5 2 ∘ sin 6 8 ∘ − sin 4 7 ∘ cos 7 7 ∘ − cos 6 5 ∘ cos 8 1 ∘ = 2 1 ​

Tanpa menggunakan tabel trigonometri, buktikan:

 

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Mahasiswa/Alumni Universitas Sriwijaya

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terbukti bahwa .

terbukti bahwa sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree equals 1 half.

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Ingat kembali: Akan dibuktikan . Maka: Jadi, terbukti bahwa .

Ingat kembali:

2 space sin space straight alpha space cos space straight beta equals sin space open parentheses straight alpha plus straight beta close parentheses plus sin space open parentheses straight alpha minus straight beta close parentheses 2 space cos space straight alpha space cos space straight beta equals cos space open parentheses straight alpha plus straight beta close parentheses minus cos space open parentheses straight alpha minus straight beta close parentheses 2 space sin space straight alpha space sin space straight beta equals negative cos space open parentheses straight alpha plus straight beta close parentheses plus cos space open parentheses straight alpha minus straight beta close parentheses sin space open parentheses 180 degree minus straight A close parentheses equals sin space straight A cos space open parentheses 90 degree plus straight A close parentheses equals negative sin space straight A 

Akan dibuktikan sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree equals 1 half. Maka:

 begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell Ruas space kiri end cell equals cell sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree end cell row blank equals cell 1 half times 2 times sin space 52 degree space sin space 68 degree minus 1 half times 2 times sin space 47 degree space cos space 77 degree minus 1 half times 2 times cos space 65 degree space cos space 81 degree end cell row blank equals cell 1 half times open parentheses 2 times sin space 52 degree space sin space 68 degree close parentheses minus 1 half times open parentheses 2 times sin space 47 degree space cos space 77 degree close parentheses minus 1 half times open parentheses 2 times cos space 65 degree space cos space 81 degree close parentheses end cell row blank equals cell 1 half open square brackets negative cos space open parentheses 52 degree plus 68 degree close parentheses plus cos space open parentheses 52 degree minus 68 degree close parentheses close square brackets minus 1 half open square brackets sin space open parentheses 47 degree plus 77 degree close parentheses plus sin space open parentheses 47 degree minus 77 degree close parentheses close square brackets minus 1 half open square brackets cos space open parentheses 65 degree plus 81 degree close parentheses plus cos space open parentheses 65 degree minus 81 degree close parentheses close square brackets end cell row blank equals cell 1 half open square brackets negative cos space 120 degree plus cos space open parentheses negative 16 degree close parentheses close square brackets minus 1 half open square brackets sin space 124 degree plus sin space open parentheses negative 30 degree close parentheses close square brackets minus 1 half open square brackets cos space 146 degree plus cos space open parentheses negative 16 degree close parentheses close square brackets end cell row blank equals cell 1 half open square brackets negative open parentheses negative 1 half close parentheses minus cos space 16 degree close square brackets minus 1 half open square brackets sin space 124 degree minus sin space 30 degree close square brackets minus 1 half open square brackets cos space 146 degree minus cos space 16 degree close square brackets end cell row blank equals cell 1 half open square brackets 1 half minus cos space 16 degree close square brackets minus 1 half open square brackets sin space 124 degree minus 1 half close square brackets minus 1 half open square brackets cos space 146 degree minus cos space 16 degree close square brackets end cell row blank equals cell 1 half open parentheses 1 half up diagonal strike negative cos space 16 degree end strike minus sin space 124 degree plus 1 half minus cos space 146 degree up diagonal strike plus cos space 16 end strike degree close parentheses end cell row blank equals cell 1 half open parentheses 1 minus sin space 124 degree minus cos space 146 degree close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses sin space 124 degree plus cos space 146 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses sin space open parentheses 180 degree minus 56 degree close parentheses plus cos space open parentheses 90 degree plus 56 degree close parentheses close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus open parentheses up diagonal strike sin space 56 degree end strike up diagonal strike negative sin space 56 degree end strike close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 minus 0 close parentheses end cell row blank equals cell 1 half open parentheses 1 close parentheses end cell row blank equals cell 1 half end cell end table end style

Jadi, terbukti bahwa sin space 52 degree space sin space 68 degree minus sin space 47 degree space cos space 77 degree minus cos space 65 degree space cos space 81 degree equals 1 half.

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