Question

Himpunan Penyelesaian Trigonometri​

Answer 1

TRiGoNoMeTRi

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tan (2x + ¹/₃π) = ¹/₃√3

tan (2x + ¹/₃π) = tan ¹/₃π

       2x + ¹/₃π = ¹/₃π

                 2x = 0

x = α + k . π → rumus

2x = 0π + k . π

 x = 1/2 kπ

k = 0 → x = 0π

k = 1  → x = 1/2 π

k = 2 → x = π

k = 3 → x = 3/2π

k = 4 → x = 2π

HP = {0π , 1/2π , π , 3/2 π , 2π}

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2 cos² 3x - cos 3x - 1 = 0

misal p = cos 3x

2p² - p - 1 = 0

(2p + 1)(p - 1) = 0

maka p = -1/2 ∨ p = 1

jika p = -1/2

cos 3x = cos 2/3 π

      3x = 2/3 π

x = α + k . 2π → rumus

3x = 2/3 π + k .  2π

  x = 2/9 π + 2/3 kπ

k = 0 → x = 2/9 π

k = 1  → x = 8/9 π

k = 2 → x = 14/9 π

x = -α + k . 2π → rumus

3x = -2/3 π + k .  2π

  x = -2/9 π + 2/3 kπ

k = 1 → x = 4/9 π

k = 2 → x = 10/9 π

k = 3 → x = 16/9 π

jika p = 1 , maka :

cos 3x = 1

cos 3x = cos 0

      3x = 0

x = α + k . 2π → rumus

3x = 0π + k .  2π

 x = 2/3 kπ

k = 0 → x = 0π

k = 1  → x = 2/3 π

k = 2 → x = 4/3 π

k = 3 → x = 2π

maka,

HP = {0π , 2/9 π , 4/9 π , 6/9 π , 8/9 π , 10/9 π, 12/9 π , 14/9 π, 16/9 π, 2π}

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Answer 2

Verified answer

PEMBAHASAN

Trigonometri

soal pertama

tan (2x + 1/3 π) = 1/3 √3

tan (2x + 1/3 π) = tan (π/6 + kπ)

2x + 1/3 π = π/6 + kπ

2x = π/6 - π/3 + kπ

2x = -π/6 + kπ

x = -π/12 + kπ/2 ... (1)

k bilangan bulat

k = 0 → x = -π/12

k = 1 → x = -π/12 + π/2 = 5π/12

k = 2 → x = -π/12 + π = 11π/12

k = 3 → x = -π/12 + 3π/2 = 17π/12

k = 4 → x = -π/12 + 2π = 23π/12

dan seterusnya

Interval 0 ≤ x ≤ 2π

HP = {5π/12 , 11π/12 , 17π/12, 23π/12}

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soal kedua

2 cos² 3x - cos 3x - 1 = 0

(2 cos 3x + 1)(cos 3x - 1) = 0

cos 3x = -1/2 atau cos 3x = 1

Interval 0° ≤ x ≤ 360°

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cos 3x = 1

3x = 0° + k.360°

x = 120°k

k bilangan bulat

x = {0° , 120° , 240°, 360°}

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cos 3x = -1/2

3x = 120° + k.360°

x = 40° + k.120° ... (1)

3x = 240° + k.360°

x = 80° + k.120° ... (2)

x = {40°, 80°, 160°, 200°, 280°, 320°}

HP = {0°, 40°, 80°, 120°, 160°, 200°, 240°, 280°, 320°, 360°}