Pertanyaan

Tentukan himpunan penyelesaian dari persamaan-persamaan trigonometri berikut! sin 4 x = sin 4 8 ∘ , 0 ∘ ≤ x ≤ 36 0 ∘

Tentukan himpunan penyelesaian dari persamaan-persamaan trigonometri berikut!

$sin 4 x = sin 4 8^{\circ}, 0^{\circ} \leq x \leq 36 0^{\circ}$

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

HP = { 1 2 ∘ , 3 3 ∘ , 10 2 ∘ , 12 3 ∘ , 19 2 ∘ , 21 3 ∘ , 28 2 ∘ , 30 3 ∘ } .

HP = {1 2^{\circ}, 3 3^{\circ}, 10 2^{\circ}, 12 3^{\circ}, 19 2^{\circ}, 21 3^{\circ}, 28 2^{\circ}, 30 3^{\circ}}

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah . Ingat jika sin x = sin α , maka x = α + k ⋅ 36 0 ∘ atau x = ( 18 0 ∘ − α ) + k ⋅ 36 0 ∘ Diketahui sin 4 x = sin 4 8 ∘ untuk 0 ∘ ≤ x ≤ 36 0 ∘ , maka berdasarkan rumus dapat ditentukan: untuk k bilangan bulat maka, k k k k k = = = = = 0 , x = 1 2 ∘ + 0 ⋅ 9 0 ∘ = 1 2 ∘ ( memenuhi ) 1 , x = 1 2 ∘ + 1 ⋅ 9 0 ∘ = 10 2 ∘ ( memenuhi ) 2 , x = 1 2 ∘ + 2 ⋅ 9 0 ∘ = 19 2 ∘ ( memenuhi ) 3 , x = 1 2 ∘ + 3 ⋅ 9 0 ∘ = 28 2 ∘ ( memenuhi ) 4 , x = 1 2 ∘ + 4 ⋅ 9 0 ∘ = 37 2 ∘ ( tidak memenuhi ) untuk k bilangan bulat maka, k k k k k = = = = = 0 , x = 3 3 ∘ + 0 ⋅ 9 0 ∘ = 3 3 ∘ ( memenuhi ) 1 , x = 3 3 ∘ + 1 ⋅ 9 0 ∘ = 12 3 ∘ ( memenuhi ) 2 , x = 3 3 ∘ + 2 ⋅ 9 0 ∘ = 21 3 ∘ ( memenuhi ) 3 , x = 3 3 ∘ + 3 ⋅ 9 0 ∘ = 30 3 ∘ ( memenuhi ) 4 , x = 3 3 ∘ + 4 ⋅ 9 0 ∘ = 39 3 ∘ ( tidak memenuhi ) Dengan demikian, HP = { 1 2 ∘ , 3 3 ∘ , 10 2 ∘ , 12 3 ∘ , 19 2 ∘ , 21 3 ∘ , 28 2 ∘ , 30 3 ∘ } .

Jawaban yang benar untuk pertanyaan tersebut adalah $\begin{array}{rcl}\mathbf{HP}&\boldsymbol=&{\boldsymbol\{{\mathbf{12}\boldsymbol\textdegree\boldsymbol,\boldsymbol\;\mathbf{33}\boldsymbol\textdegree\boldsymbol,\boldsymbol\;\mathbf{102}\boldsymbol\textdegree\boldsymbol,\boldsymbol\;\mathbf{123}\boldsymbol\textdegree\boldsymbol,\boldsymbol\;\mathbf{192}\boldsymbol\textdegree\boldsymbol,\boldsymbol\;\mathbf{213}\boldsymbol\textdegree\boldsymbol,\boldsymbol\;\mathbf{282}\boldsymbol\textdegree\boldsymbol,\boldsymbol\;\mathbf{303}\boldsymbol\textdegree\boldsymbol\}}}\end{array}$ .

Ingat jika $sin x = sin α$ , maka

$x = α + k \cdot 36 0^{\circ}$ atau
$x = (18 0^{\circ} - α) + k \cdot 36 0^{\circ}$

Diketahui $sin 4 x = sin 4 8^{\circ}$ untuk $0^{\circ} \leq x \leq 36 0^{\circ}$ , maka berdasarkan rumus dapat ditentukan:

$\begin{array}{rcl}\mathbf4\boldsymbol x&\boldsymbol=&\mathbf{48}\boldsymbol\textdegree\boldsymbol+\boldsymbol k\boldsymbol\cdot\mathbf{360}\boldsymbol\textdegree\\x&=&12^\circ+k\cdot90^\circ\end{array}$

untuk $k$ bilangan bulat maka,

$k k k k k = = = = = 0, x = 1 2^{\circ} + 0 \cdot 9 0^{\circ} = 1 2^{\circ} (memenuhi) 1, x = 1 2^{\circ} + 1 \cdot 9 0^{\circ} = 10 2^{\circ} (memenuhi) 2, x = 1 2^{\circ} + 2 \cdot 9 0^{\circ} = 19 2^{\circ} (memenuhi) 3, x = 1 2^{\circ} + 3 \cdot 9 0^{\circ} = 28 2^{\circ} (memenuhi) 4, x = 1 2^{\circ} + 4 \cdot 9 0^{\circ} = 37 2^{\circ} (tidak memenuhi)$

$\begin{array}{rcl}\mathbf4\boldsymbol x&\boldsymbol=&\boldsymbol(\mathbf{180}\boldsymbol-\mathbf{48}\boldsymbol\textdegree\boldsymbol)\boldsymbol+\boldsymbol k\boldsymbol\cdot\mathbf{360}\boldsymbol\textdegree\\4x&=&132^\circ+k\cdot360^\circ\\x&=&33^\circ+k\cdot90^\circ\end{array}$

untuk $k$ bilangan bulat maka,

$k k k k k = = = = = 0, x = 3 3^{\circ} + 0 \cdot 9 0^{\circ} = 3 3^{\circ} (memenuhi) 1, x = 3 3^{\circ} + 1 \cdot 9 0^{\circ} = 12 3^{\circ} (memenuhi) 2, x = 3 3^{\circ} + 2 \cdot 9 0^{\circ} = 21 3^{\circ} (memenuhi) 3, x = 3 3^{\circ} + 3 \cdot 9 0^{\circ} = 30 3^{\circ} (memenuhi) 4, x = 3 3^{\circ} + 4 \cdot 9 0^{\circ} = 39 3^{\circ} (tidak memenuhi)$