tugas kelompok matmin trigonometri 11 mipa 5

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TUGAS KELOMPOK MATEMATIKA PEMINATAN PERSAMAAN TRIGONOMETRI

NAMA ANGGOTA KELOMPOK ADINA KHAIRUNISA (01) AMNA NUZILA ASIFA (04)

ANTARIKA ALYA HAWANINGSIH (05) DINI AMIARTI (07)

FAZA ARDIAN.M (12) FERA KIRANI (13) SOAL

1. Tentukan himpunan penyelesaian dari

cos❑ x = 0 untuk 0° ≤ x ≤ 360°

6 sin❑ 6x =0 untuk 90° ≤ x ≤ 360°

3. Tentukan himpunan penyelesaian dari cos❑ (3x-60°) = −1

2 untuk 0° ≤ x ≤ 360°

2 sin²❑+¿ 4 ^sin❑x –6 = 0 untuk 0° ^≤ x ^≤ 360°

5. Tentukan himpunan penyelesaian dari ^cos❑ 5x = 1

2 untuk 0° ^≤ x ^≤ 180°

2 sin²❑−𝑥−9 cos❑𝑥+3 = 0 Untuk 0°≤ x ≤ 360°

7 . Tentukan himpunan penyelesaian dari persamaan tan❑ (2x-1

4 𝜋) = 1

4𝜋 untuk 0≤ x ≤2𝜋

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PEMBAHASAN

1. cos❑ x = 0 cos❑ x = cos❑ 90°

∴ ^cos❑ x = ∝+ k.360°

x = 90° + k.360°

K= 0 ^→x = 90° + 0.360°

x = 90° + 0 ° x = 90°

K = 1 ^→x = 90° + 1.360°

x = 90° + 360°

x = 450° (TM)

∴^cos❑ x = -∝+ k.360°

X = -90° + k.360°

K = 0 →x = -90° + 0.360°

x = -90° + 0°

x = -90°

K = 1 →x = -90° + 1.360°

x = -90° + 360°

x = 270°

K = 2 →x = -90°+ 2.360°

x = -90° + 720°

x = 630° (TM) HP = { 90° , 270 ° )

2. 6 sin❑ 6x = 0

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6 sin❑ 6x = sin❑ 0°

sin❑ 6x = sin❑ 0° : 6 sin❑ 6x = sin❑ 0°

∴^sin❑ x = ∝ + k.360°

sin❑ 6x = 0° + k.360°

sin❑ x =( 0° + k.360°) : 6 ^sin❑ x = 0° + k. 60°

K = 0 →x = 0° + 0.60°

x = 0° (TM)

K = 1 ^→x = 0° + 1.60°

x = 0°+ 60°

x = 60° (TM)

K = 2 →x = 0°+ 2.60°

x = 0° + 120°

x = 120°

K = 3 →x = 0° + 3.60°

x = 0° + 180°

x = 180°

K= 4 →x = 0° + 4.60°

x = 0° + 240°

x = 240°

K = 5 → x = 0° + 5.60°

x = 0° + 300°

x = 300°

K = 6 → x = 0° + 6.60°

x = 0° + 360°

x = 360°

K = 7 →x = 0° + 7.60°

x = 0° + 420°

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x = 420° (TM)

∴sin❑ x = ( 180°-∝) + k.360°

sin❑ 6x =( 180°- 0°) + k.360°

6x = 180°+ k.360°

x = ( 180° + k.360°): 6 x = 30°+ k.60°

K = 0→x = 30°+ 0.60°

x = 30° + 0°

x = 30° (TM)

K = 1 →x = 30°+ 1.60°

x = 30°+ 120°

x= 90°

K= 2→x = 30°+ 2.60°

x = 30° + 120°

x = 150°

K = 3 →x = 30°+ 3.60°

x = 30°+ 180°

x = 210°

K = 4→x = 30° + 4.60°

x = 30°+ 240°

x = 270°

K = 5 ^→x = 30° + 5.60°

x = 30° + 300°

x = 330°

K = 6^→x = 30° + 6.60°

x = 30° + 360°

x = 390° (TM)

HP = { 90°, 120 ° , 180°,210° , 270° , 300° , 330°, 360° }

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3. ^cos❑ ( 3x-60°) = −1 2 cos❑ ( 3x-60°) = cos❑ 120°

∴^cos❑ x =∝+ k.360°

cos❑ (3x-60°) = 120° + k.360°

3x = ( 120°+ 60°) + k.360°

3x = 180°+k.360°

x = ( 180° + k.360°) : 3 x = 60° + k.120°

K= 0 ^→x = 60° + 0.120°

x = 60° + 0°

x = 60°

K= 1 → x = 60° + 1.120°

x = 60° + 120°

x = 180°

K=2 →x = 60° + 2.120°

x = 60° + 240°

x = 300°

K = 3 →x = 60° + 3.120°

x = 60° + 360°

x = 420° (TM)

∴cos❑ x = −∝+ k.360°

cos❑ (3x-60°) = -120° +k.360°

3x =( -120°+60°) + k.360°

3x = -60° + k.360°

x = (-60° + k.360°) : 3 x = - 20° + k.120°

K = 0 →x = -20°+ 0.120°

x = -20°+0°

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x = - 20° (TM)

K = 1 →x = -20° + 1.120°

x = -20° + 120°

x = 100°

K = 2 →x = -20° + 2.120°

x = - 20° + 240°

x = 220°

K = 3 →x = -20° + 3.120°

x = - 20° + 360°

x = 340°

K = 4 →x = - 20° + 4.120°

x = -20° + 480°

x = 460° (TM)

HP = { 60° , 100° , 180° , 220° , 300° , 340° }

4. ∴Misal sin❑ x = a,maka 2a²+ 4a – 6 = 0

(2a-2)(a+3) = 0

2a-2 = 0 atau a+3 = 0

a = 2

2 atau a= -3 a = 1 atau a= -3

∴Ingat ! -1 ≤sin❑ 2 ≤ 1, maka nilai a yang memenuhi adalah 1 ,maka A = 1

sin❑ x = 1 sin❑ x = sin❑ 90°

∴X = ∝+ k . 360^° atau x = - ∝+ k. 360^° X = 90°+ k. 360° atau x = -90°+ k. 360°

∴X = 90°+ k. 360° K = 0 ^→x = 90^°+ 0. 360^°

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x = 90° K = 1 →x = 90°+ 1. 360°

x = 90°+ 360° x = 450°(TM)

∴X = -90°+ k. 360° K = 0 →x = -90° + 0. 360°

x = -90° + 0°

x = -90° (TM)

K = 1→x = - 90° + 1.360°

x = - 90° + 360°

x = 270°

K = 2→x = -90° + 2.360°

x = - 90° + 720°

x = 630°

HP = { 90° , 270° }

5. cos❑ 5x = 1 2 cos❑ 5x = cos❑ 60°

∴ ^cos❑ x =∝+ k.360°

cos❑ 5x = 60° + k.360°

x = (60° + k.360°) : 5 x = 12° + k.72°

K = 0 →x = 12° +0.72°

x = 12° + 0°

x = 12°

K = 1 ^→x = 12° + 1.72°

x = 12° + 72°

x = 84°

K = 2 ^→x = 12° + 2.72°

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x = 12° + 144°

x = 156°

K = 3 →x = 12° + 3.72°

x = 12° + 216°

x = 228° (TM)

∴^cos❑ x = −∝ + k.360°

cos❑ 5x = - 60° + k.360°

cos❑ x = (-60°+ k.360°) : 5 X = - 12° + k.72°

K = 0 →x = -12° + 0.72°

x = - 12° + 0°

x = - 12° (TM)

K = 1 →x = - 12° + 1.72°

x = -12° + 72°

x = 60°

K = 2 →x = -12° + 2.72°

x = - 12° + 144°

x = 132°

K = 3 →x = - 12° + 3.72 ° x = - 12° + 216°

x = 204° (TM)

HP = { 12° , 60° , 84° , 132°, 156° , }

6. 2 sin²❑𝑥−9 cos❑𝑥+3=0 2

(

^cos²❑𝑥

)

⁻^{9 cos}❑𝑥+3=0 2−2 cos²❑𝑥−9 cos❑𝑥+3=0

2 cos²❑𝑥+9 cos❑𝑥−5=0 (2 cos❑𝑥−1) (cos❑𝑥+5)=0 2 cos❑𝑥−1=0 atau cos❑𝑥+5=0 2 cos❑𝑥=1 atau cos❑𝑥=−5 (TM)

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cos❑𝑥=1 2 cos❑𝑥=60° ∴^cos❑𝑥=¿∝ + k.360°

𝑥 = 60° + k .360°

K = 0 →x = 60° + 0.360°

X = 60 ° + 0°

X = 60°

K = 1 →x = 60° + 1.360°

X = 60° + 360°

X = 420° (TM)

∴cos❑ x = -∝+ k.360°

cos❑ x = - 60° + k.360°

K = 0 →x = -60° + 0.360°

X = - 60°+ 0°

X = -60° (TM)

K = 1 →x = -60° + 1.360°

X = -60° + 360°

X = 300°

K = 2 →x = - 60° + 2.360°

X = -60° + 720°

X = 660° (TM) HP = { 60° , 300° }

7.tan❑ x =∝+ k .180°

tan❑ ( 2x - 1

4 𝜋) = 1

4𝜋 + k.𝜋 tan❑ 2x =

(

¹⁴^𝜋⁺¹⁴ ^𝜋

)

⁺^𝑘^.^𝜋

2x =2

4𝜋+𝑘.𝜋

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x =

(

²⁴ ^𝜋⁺^𝑘^.^𝜋

)

^{: 2}

x =1

4𝜋+𝑘.1 2𝜋 K =0 → x =1

4𝜋+𝑘.1 2𝜋 x = 1

4𝜋 K = 1 →x =1

4𝜋+1.1 2𝜋 x =1

4 𝜋+1 2𝜋 x = 3

4𝜋 K = 2 →x = 1

4𝜋+2.1 2𝜋 x = 1

4 𝜋+1 1𝜋 x = 5

4𝜋 K = 3→x =1

4𝜋+3.1 2𝜋 x =1

4 𝜋+3 2𝜋 x =7

4𝜋 K = 4 →x = 1

4𝜋+4.1 2𝜋 x = 1

4 𝜋+2𝜋 x = 9

4 𝜋 ( TM) HP =

{

¹⁴ ^𝜋^,³⁴^𝜋^,⁵⁴ ^𝜋^,⁷⁴

}

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