Pertanyaan

Tentukan nilai x yang memenuhi persamaan berikut untuk 0 ∘ ≤ x ≤ 36 0 ∘ . b. ( cot ( 2 x − 60 ) ∘ + 3 ) ( sec 3 x − 2 ) = 0

Tentukan nilai $x$ yang memenuhi persamaan berikut untuk $0^{\circ} \leq x \leq 36 0^{\circ}$ .

b. $(cot (2 x - 60)^{\circ} + 3) (sec 3 x - 2) = 0$

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A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

nilai x yang memenuhi persamaan ( cot ( 2 x − 60 ) ∘ + 3 ) ( sec 3 x − 2 ) = 0 untuk 0 ∘ ≤ x ≤ 36 0 ∘ adalah { 1 5 ∘ , 10 5 ∘ , 13 5 ∘ , 19 5 ∘ , 22 5 ∘ , 25 5 ∘ , 28 5 ∘ , 34 5 ∘ } .

nilai

x

yang memenuhi persamaan

(cot (2 x - 60)^{\circ} + 3) (sec 3 x - 2) = 0

untuk

0^{\circ} \leq x \leq 36 0^{\circ}

adalah

{1 5^{\circ}, 10 5^{\circ}, 13 5^{\circ}, 19 5^{\circ}, 22 5^{\circ}, 25 5^{\circ}, 28 5^{\circ}, 34 5^{\circ}}

Pembahasan

Ingat kembali: cot x = tan x 1 sec x = cos x 1 cos x = cos α { x = α + k ⋅ 36 0 ∘ x = − α + k ⋅ 36 0 ∘ tan x = tan α → x = α + k ⋅ 18 0 ∘ Diberikan persamaan ( cot ( 2 x − 60 ) ∘ + 3 ) ( sec 3 x − 2 ) = 0 untuk 0 ∘ ≤ x ≤ 36 0 ∘ . Maka: - Untuk ( cot ( 2 x − 60 ) ∘ + 3 ) = 0 ( cot ( 2 x − 60 ) ∘ + 3 ) = 0 cot ( 2 x − 60 ) ∘ + 3 = 0 cot ( 2 x − 60 ) ∘ = − 3 tan ( 2 x − 60 ) ∘ 1 = − 3 − 3 1 = tan ( 2 x − 60 ) ∘ tan ( 2 x − 60 ) ∘ = − 3 1 tan ( 2 x − 60 ) ∘ = tan 15 0 ∘ Persamaan tangen 2 x − 6 0 ∘ 2 x x = = = 15 0 ∘ + k ⋅ 18 0 ∘ 21 0 ∘ + k ⋅ 18 0 ∘ 10 5 ∘ + k ⋅ 9 0 ∘ - Untuk ( sec 3 x − 2 ) = 0 ( sec 3 x − 2 ) sec 3 x − 2 sec 3 x c o s 3 x 1 2 1 cos 3 x cos 3 x cos 3 x = = = = = = = = 0 0 2 2 cos 3 x 2 1 2 1 2 cos 4 5 ∘ Persamaan cosinus (1) 3 x x = = 4 5 ∘ + k ⋅ 36 0 ∘ 1 5 ∘ + k ⋅ 12 0 ∘ k = 0 → x = 1 5 ∘ + 0 ⋅ 12 0 ∘ = 1 5 ∘ + 0 ∘ = 1 5 ∘ k = 1 → x = 1 5 ∘ + 1 ⋅ 12 0 ∘ = 1 5 ∘ + 12 0 ∘ = 13 5 ∘ k = 2 → x = 1 5 ∘ + 2 ⋅ 12 0 ∘ = 1 5 ∘ + 24 0 ∘ = 25 5 ∘ Persamaan cosinus (2) 3 x x = = − 4 5 ∘ + k ⋅ 36 0 ∘ − 1 5 ∘ + k ⋅ 12 0 ∘ k = 1 → x = − 1 5 ∘ + 1 ⋅ 12 0 ∘ = − 1 5 ∘ + 12 0 ∘ = 10 5 ∘ k = 2 → x = − 1 5 ∘ + 2 ⋅ 12 0 ∘ = − 1 5 ∘ + 24 0 ∘ = 22 5 ∘ k = 3 → x = − 1 5 ∘ + 3 ⋅ 12 0 ∘ = − 1 5 ∘ + 36 0 ∘ = 34 5 ∘ Jadi,nilai x yang memenuhi persamaan ( cot ( 2 x − 60 ) ∘ + 3 ) ( sec 3 x − 2 ) = 0 untuk 0 ∘ ≤ x ≤ 36 0 ∘ adalah { 1 5 ∘ , 10 5 ∘ , 13 5 ∘ , 19 5 ∘ , 22 5 ∘ , 25 5 ∘ , 28 5 ∘ , 34 5 ∘ } .

Ingat kembali:

$cot x = \frac{1}{tan x}$
$sec x = \frac{1}{cos x}$
$cos x = cos α {x = α + k \cdot 36 0^{\circ} x = - α + k \cdot 36 0^{\circ}$
$tan x = tan α \to x = α + k \cdot 18 0^{\circ}$

Diberikan persamaan $(cot (2 x - 60)^{\circ} + 3) (sec 3 x - 2) = 0$ untuk $0^{\circ} \leq x \leq 36 0^{\circ}$ . Maka:

- Untuk $(cot (2 x - 60)^{\circ} + 3) = 0$

$(cot (2 x - 60)^{\circ} + 3) = 0 cot (2 x - 60)^{\circ} + 3 = 0 cot (2 x - 60)^{\circ} = - 3 \frac{1}{tan ( 2 x - 60 ) ^{\circ}} = - 3 \frac{1}{- 3} = tan (2 x - 60)^{\circ} tan (2 x - 60)^{\circ} = \frac{1}{- 3} tan (2 x - 60)^{\circ} = tan 15 0^{\circ}$

Persamaan tangen

$2 x - 6 0^{\circ} 2 x x = = = 15 0^{\circ} + k \cdot 18 0^{\circ} 21 0^{\circ} + k \cdot 18 0^{\circ} 10 5^{\circ} + k \cdot 9 0^{\circ}$

$k=0\rightarrow\begin{array}{rcl}&&x\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&105\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&&+\end{array}\begin{array}{rcl}&&0\end{array}\begin{array}{rcl}&&\cdot\end{array}\begin{array}{rcl}&&90\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&105\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&&+\end{array}\begin{array}{rcl}&&0\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&105\end{array}\begin{array}{rcl}&&\textdegree\end{array}\\\begin{array}{rcl}&&k\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&1\end{array}\begin{array}{rcl}&\rightarrow&\end{array}\begin{array}{rcl}&&x\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&105\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&&+\end{array}\begin{array}{rcl}&&1\end{array}\begin{array}{rcl}&&\cdot\end{array}\begin{array}{rcl}&&90\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&105\end{array}\begin{array}{rcl}\textdegree+90^\circ&=&195^\circ\end{array}\\\begin{array}{rcl}&&k\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&2\end{array}\begin{array}{rcl}&\rightarrow&\end{array}\begin{array}{rcl}&&x\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&105\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&&+\end{array}\begin{array}{rcl}&&2\end{array}\begin{array}{rcl}&&\cdot\end{array}\begin{array}{rcl}&&90\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&105\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&&+\end{array}\begin{array}{rcl}&&180\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&285\end{array}\begin{array}{rcl}&&\textdegree\end{array}$

- Untuk $(sec 3 x - 2) = 0$

$(sec 3 x - 2) sec 3 x - 2 sec 3 x \frac{1}{c o s 3 x} \frac{1}{2} cos 3 x cos 3 x cos 3 x = = = = = = = = 0022 cos 3 x \frac{1}{2} \frac{1}{2} 2 cos 4 5^{\circ}$

Persamaan cosinus (1)

$3 x x = = 4 5^{\circ} + k \cdot 36 0^{\circ} 1 5^{\circ} + k \cdot 12 0^{\circ}$

$k = 0 \to x = 1 5^{\circ} + 0 \cdot 12 0^{\circ} = 1 5^{\circ} + 0^{\circ} = 1 5^{\circ} k = 1 \to x = 1 5^{\circ} + 1 \cdot 12 0^{\circ} = 1 5^{\circ} + 12 0^{\circ} = 13 5^{\circ} k = 2 \to x = 1 5^{\circ} + 2 \cdot 12 0^{\circ} = 1 5^{\circ} + 24 0^{\circ} = 25 5^{\circ}$

Persamaan cosinus (2)

$3 x x = = - 4 5^{\circ} + k \cdot 36 0^{\circ} - 1 5^{\circ} + k \cdot 12 0^{\circ}$

$k = 1 \to x = - 1 5^{\circ} + 1 \cdot 12 0^{\circ} = - 1 5^{\circ} + 12 0^{\circ} = 10 5^{\circ} k = 2 \to x = - 1 5^{\circ} + 2 \cdot 12 0^{\circ} = - 1 5^{\circ} + 24 0^{\circ} = 22 5^{\circ} k = 3 \to x = - 1 5^{\circ} + 3 \cdot 12 0^{\circ} = - 1 5^{\circ} + 36 0^{\circ} = 34 5^{\circ}$

Jadi, nilai $x$ yang memenuhi persamaan $(cot (2 x - 60)^{\circ} + 3) (sec 3 x - 2) = 0$ untuk $0^{\circ} \leq x \leq 36 0^{\circ}$ adalah ${1 5^{\circ}, 10 5^{\circ}, 13 5^{\circ}, 19 5^{\circ}, 22 5^{\circ}, 25 5^{\circ}, 28 5^{\circ}, 34 5^{\circ}}$ .

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