Pertanyaan

Jika tan 2 5 ∘ = a , hitunglah: a. 1 + tan 15 5 ∘ ⋅ tan 11 5 ∘ tan 15 5 ∘ − tan 11 5 ∘

Jika $tan 2 5^{\circ} = a$ , hitunglah:

a. $\frac{tan 15 5 ^{\circ} - tan 11 5 ^{\circ}}{1 + tan 15 5 ^{\circ} \cdot tan 11 5 ^{\circ}}$

S. Nur

Master Teacher

Jawaban terverifikasi

Jawaban

diperoleh

diperoleh $fraction numerator tan space 155 degree minus tan space 115 degree over denominator 1 plus tan space 155 degree times tan space 115 degree end fraction equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator size 12px minus size 12px a to the power of size 12px 2 size 12px plus size 12px 1 over denominator size 12px 2 size 12px a end fraction end cell end table$

Pembahasan

Ingat rumus sudut berelasi tangen pada kuadran II: tan ( 18 0 ∘ − x ) tan ( 9 0 ∘ + x ) = = − tan x − cot x = − t a n x 1 Maka, 1 + t a n 15 5 ∘ ⋅ t a n 11 5 ∘ t a n 15 5 ∘ − t a n 11 5 ∘ = = = = = 1 + t a n ( 180 − 25 ) ∘ ⋅ t a n ( 90 + 25 ) ∘ t a n ( 180 − 25 ) ∘ − t a n ( 90 + 25 ) ∘ 1 + ( − t a n 2 5 ∘ ) ( − c o t 2 5 ∘ ) − t a n 2 5 ∘ − ( − c o t 2 5 ∘ ) 1 + t a n 2 5 ∘ ⋅ tan 2 5 ∘ 1 − t a n 2 5 ∘ + tan 2 5 ∘ 1 1 + 1 − a + a 1 2 a − a 2 + 1 Jadi, diperoleh

Ingat rumus sudut berelasi tangen pada kuadran II:

$tan (18 0^{\circ} - x) tan (9 0^{\circ} + x) = = - tan x - cot x = - \frac{1}{t a n x}$

Maka,

$\frac{t a n 15 5 ^{\circ} - t a n 11 5 ^{\circ}}{1 + t a n 15 5 ^{\circ} \cdot t a n 11 5 ^{\circ}} = = = = = \frac{t a n ( 180 - 25 ) ^{\circ} - t a n ( 90 + 25 ) ^{\circ}}{1 + t a n ( 180 - 25 ) ^{\circ} \cdot t a n ( 90 + 25 ) ^{\circ}} \frac{- t a n 2 5 ^{\circ} - ( - c o t 2 5 ^{\circ} )}{1 + ( - t a n 2 5 ^{\circ} ) ( - c o t 2 5 ^{\circ} )} \frac{- t a n 2 5 ^{\circ} + \frac{1}{tan 2 5 ^{\circ}}}{1 + t a n 2 5 ^{\circ} \cdot \frac{1}{tan 2 5 ^{\circ}}} \frac{- a + \frac{1}{a}}{1 + 1} \frac{- a ^{2} + 1}{2 a}$

Jadi, diperoleh $fraction numerator tan space 155 degree minus tan space 115 degree over denominator 1 plus tan space 155 degree times tan space 115 degree end fraction equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator size 12px minus size 12px a to the power of size 12px 2 size 12px plus size 12px 1 over denominator size 12px 2 size 12px a end fraction end cell end table$