Pertanyaan

Nilai eksak dari ekspresi trigonometri: tan 5 ∘ ⋅ tan 2 5 ∘ ⋅ tan 4 5 ∘ ⋅ tan 6 5 ∘ ⋅ tan 8 5 ∘ adalah . . . .

Nilai eksak dari ekspresi trigonometri: $tan 5^{\circ} \cdot tan 2 5^{\circ} \cdot tan 4 5^{\circ} \cdot tan 6 5^{\circ} \cdot tan 8 5^{\circ}$ adalah . . . .

$- 1$
$- \frac{1}{2}$
$0$
$\frac{1}{2}$
$1$

L. Nikmah

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah E.

Pembahasan

Ingat identitas trigonometri tan x = cos x sin x dan rumus perkalian trigonometri berikut ini: − 2 sin A sin B = cos ( A + B ) − cos ( A − B ) 2 cos A cos B = cos ( A + B ) + cos ( A − B ) Dengan menggunakan konsep tersebut, diperoleh hasil: Jadi, tan 5 ∘ ⋅ tan 2 5 ∘ ⋅ tan 4 5 ∘ ⋅ tan 6 5 ∘ ⋅ tan 8 5 ∘ = 1 . Oleh karena itu, jawaban yang benar adalah E.

Ingat identitas trigonometri $tan x = \frac{sin x}{cos x}$ dan rumus perkalian trigonometri berikut ini:

$- 2 sin A sin B = cos (A + B) - cos (A - B)$

$2 cos A cos B = cos (A + B) + cos (A - B)$

Dengan menggunakan konsep tersebut, diperoleh hasil:

$tan space 5 degree times tan space 25 degree times tan space 45 degree times tan space 65 degree times tan space 85 degree equals fraction numerator sin space 5 degree over denominator cos space 5 degree end fraction times fraction numerator sin space 25 degree over denominator cos space 25 degree end fraction times tan space 45 degree times fraction numerator sin space 65 degree over denominator cos space 65 degree end fraction times fraction numerator sin space 85 degree over denominator cos space 85 degree end fraction equals fraction numerator sin space 5 degree over denominator cos space 5 degree end fraction times fraction numerator sin space 25 degree over denominator cos space 25 degree end fraction times 1 times fraction numerator sin space 65 degree over denominator cos space 65 degree end fraction times fraction numerator sin space 85 degree over denominator cos space 85 degree end fraction equals fraction numerator sin space 5 degree over denominator cos space 5 degree end fraction times fraction numerator sin space 25 degree over denominator cos space 25 degree end fraction times fraction numerator sin space 65 degree over denominator cos space 65 degree end fraction times fraction numerator sin space 85 degree over denominator cos space 85 degree end fraction equals open parentheses fraction numerator sin space 85 degree over denominator cos space 85 degree end fraction times fraction numerator sin space 5 degree over denominator cos space 5 degree end fraction close parentheses open parentheses fraction numerator sin space 65 degree over denominator cos space 65 degree end fraction times fraction numerator sin space 25 degree over denominator cos space 25 degree end fraction close parentheses equals open parentheses fraction numerator sin space 85 degree space sin space 5 degree over denominator cos space 85 degree space cos space 5 degree end fraction close parentheses open parentheses fraction numerator sin space 65 degree space sin space 25 degree over denominator cos space 65 degree space cos space 25 degree end fraction close parentheses equals open parentheses fraction numerator 1 half open parentheses negative cos space open parentheses 85 degree plus 5 degree close parentheses plus cos space open parentheses 85 degree minus 5 degree close parentheses close parentheses over denominator 1 half open parentheses cos space open parentheses 85 degree plus 5 degree close parentheses plus cos space open parentheses 85 degree minus 5 degree close parentheses close parentheses end fraction close parentheses open parentheses fraction numerator 1 half open parentheses negative cos space open parentheses 65 degree plus 25 degree close parentheses plus cos space open parentheses 65 degree minus 25 degree close parentheses close parentheses over denominator 1 half open parentheses cos space open parentheses 65 degree plus 25 degree close parentheses plus cos space open parentheses 65 degree minus 25 degree close parentheses close parentheses end fraction close parentheses equals open parentheses fraction numerator 1 half open parentheses negative cos space 90 degree plus cos space 80 degree close parentheses over denominator 1 half open parentheses cos space 90 degree plus cos space 80 degree close parentheses end fraction close parentheses open parentheses fraction numerator 1 half open parentheses negative cos space 90 degree plus cos space 40 degree close parentheses over denominator 1 half open parentheses cos space 90 degree plus cos space 40 degree close parentheses end fraction close parentheses equals open parentheses fraction numerator negative cos space 90 degree plus cos space 80 degree over denominator cos space 90 degree plus cos space 80 degree end fraction close parentheses open parentheses fraction numerator negative cos space 90 degree plus cos space 40 degree over denominator cos space 90 degree plus cos space 40 degree end fraction close parentheses equals open parentheses fraction numerator 0 plus cos space 80 degree over denominator 0 plus cos space 80 degree end fraction close parentheses open parentheses fraction numerator 0 plus cos space 40 degree over denominator 0 plus cos space 40 degree end fraction close parentheses equals open parentheses fraction numerator cos space 80 degree over denominator cos space 80 degree end fraction close parentheses open parentheses fraction numerator cos space 40 degree over denominator cos space 40 degree end fraction close parentheses equals 1 times 1 equals 1$

Jadi, $tan 5^{\circ} \cdot tan 2 5^{\circ} \cdot tan 4 5^{\circ} \cdot tan 6 5^{\circ} \cdot tan 8 5^{\circ} = 1$ .