Pertanyaan

Hitunglah: tan 6 9 ∘ + tan 3 3 ∘ − tan 3 9 ∘ − tan 3 ∘ .

Hitunglah:

$tan 6 9^{\circ} + tan 3 3^{\circ} - tan 3 9^{\circ} - tan 3^{\circ}$ .

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Pembahasan

Ingat bahwa: Mencari nilai :

Ingat bahwa:

$tan space A equals fraction numerator sin space A over denominator cos space B end fraction$
$sin space 18 degree equals fraction numerator square root of 5 minus 1 over denominator 4 end fraction$
$cos space 36 degree equals fraction numerator square root of 5 plus 1 over denominator 4 end fraction$

Mencari nilai tan space 69 degree plus tan space 33 degree minus tan space 39 degree minus tan space 3 degree :

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell tan open parentheses 69 degree close parentheses plus tan open parentheses 33 degree close parentheses minus tan open parentheses 39 degree close parentheses minus tan open parentheses 3 degree close parentheses end cell row blank equals cell tan open parentheses 69 degree close parentheses minus tan open parentheses 39 degree close parentheses plus tan open parentheses 33 degree close parentheses minus tan open parentheses 3 degree close parentheses end cell row blank equals cell open parentheses fraction numerator sin space 69 degree over denominator cos space 69 degree end fraction minus fraction numerator sin space 39 degree over denominator cos space 39 degree end fraction close parentheses plus open parentheses fraction numerator sin space 33 degree over denominator cos space 33 degree end fraction minus fraction numerator sin space 3 degree over denominator cos space 3 degree end fraction close parentheses end cell row blank equals cell open parentheses fraction numerator sin space 69 degree cos space 39 degree minus sin space 69 degree cos space 39 degree over denominator cos space 69 degree cos space 39 degree end fraction close parentheses end cell row blank blank cell plus open parentheses fraction numerator sin space 33 degree cos space 3 degree minus sin space 33 degree cos space 3 degree over denominator cos space 33 degree cos space 3 degree end fraction close parentheses end cell row blank equals cell fraction numerator sin open parentheses 69 degree minus 39 degree close parentheses over denominator cos space 69 degree cos space 39 degree end fraction plus fraction numerator sin open parentheses 33 degree minus 3 degree close parentheses over denominator cos space 33 degree cos space 3 degree end fraction end cell row blank equals cell fraction numerator sin open parentheses 30 degree close parentheses over denominator cos space 69 degree cos space 39 degree end fraction plus fraction numerator sin open parentheses 30 degree close parentheses over denominator cos space 33 degree cos space 3 degree end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 cos space 69 degree cos space 39 degree end fraction plus fraction numerator 1 over denominator 2 cos space 33 degree cos space 3 degree end fraction end cell row blank equals cell fraction numerator 1 over denominator cos space open parentheses 69 degree plus 39 degree close parentheses plus cos space open parentheses 69 degree minus 39 degree close parentheses end fraction end cell row blank blank cell plus fraction numerator 1 over denominator cos space open parentheses 33 degree plus 3 degree close parentheses plus cos space open parentheses 33 degree minus 3 degree close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator cos space open parentheses 108 degree close parentheses plus cos space open parentheses 30 degree close parentheses end fraction end cell row blank blank cell plus fraction numerator 1 over denominator cos space open parentheses 36 degree close parentheses plus cos space open parentheses 30 degree close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator negative sin space open parentheses 18 degree close parentheses plus cos space open parentheses 30 degree close parentheses end fraction end cell row blank blank cell plus fraction numerator 1 over denominator cos space open parentheses 30 degree close parentheses plus cos space open parentheses 36 degree close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator negative begin display style fraction numerator square root of 5 minus 1 over denominator 4 end fraction end style plus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style end fraction plus fraction numerator 1 over denominator begin display style fraction numerator square root of 3 over denominator 2 end fraction plus fraction numerator square root of 5 plus 1 over denominator 4 end fraction end style end fraction end cell row blank equals cell fraction numerator 4 over denominator negative square root of 5 plus 1 plus 2 square root of 3 end fraction plus fraction numerator 4 over denominator 2 square root of 3 plus square root of 5 plus 1 end fraction end cell row blank equals cell 4 open parentheses fraction numerator 1 over denominator negative square root of 5 plus 1 plus 2 square root of 3 end fraction plus fraction numerator 1 over denominator 2 square root of 3 plus square root of 5 plus 1 end fraction close parentheses end cell row blank equals cell 4 open parentheses fraction numerator open parentheses 2 square root of 3 plus 1 close parentheses plus square root of 5 minus square root of 5 plus open parentheses 1 plus 2 square root of 3 close parentheses over denominator open square brackets negative square root of 5 plus open parentheses 1 plus 2 square root of 3 close parentheses close square brackets open square brackets open parentheses 2 square root of 3 plus 1 close parentheses plus square root of 5 close square brackets end fraction close parentheses end cell row blank equals cell 4 open parentheses fraction numerator 2 open parentheses 2 square root of 3 plus 1 close parentheses over denominator open parentheses negative square root of 5 close parentheses squared plus open parentheses 1 plus 2 square root of 3 close parentheses squared end fraction close parentheses end cell row blank equals cell fraction numerator 8 open parentheses 2 square root of 3 plus 1 close parentheses over denominator 12 plus 1 plus 4 square root of 3 minus 5 end fraction end cell row blank equals cell 2 open parentheses 3 square root of 3 minus 4 close parentheses end cell row blank almost equal to cell 2 comma 392 end cell end table$