Question

Jadikan identitas trigonometri ini ke bentuk yang paling sederhana
[tex]\displaystyle \frac{\cot x-\csc x+1}{\cot x+\csc x-1}[/tex]

Answer 1

Jadikan identitas trigonometri ini ke bentuk paling sederhana

[tex]\sf{\dfrac{cot (x) - csc (x) + 1}{cot (x) + csc (x) - 1} = tan \left( \dfrac{x}{2} \right) }[/tex]

Pembahasan :

Misalnya :

[tex]\sf{\dfrac{cot (x) - csc (x) + 1}{cot (x) + csc (x) - 1} = \dfrac{a}{b}}[/tex]

Perhatikan

• cot adalah cotangen
• csc adalah cosec atau cosecan

[tex]\sf{cot (x) = \dfrac{1}{tan (x)} = \dfrac{cos (x)}{sin (x)}}[/tex]

[tex]\sf{cosec (x) = \dfrac{1}{sin (x)}}[/tex]

• Untuk lebih memudahkan dalam penyederhanaan soal, selesaian satu per satu secara berurutan

Pembilang

[tex]\sf{a = cot (x) - csc (x) + 1}[/tex]

[tex]\sf{a = cot (x) - cosec (x) + 1}[/tex]

[tex]\sf{a = \dfrac{cos (x)}{sin (x)} - \dfrac{1}{sin (x)} + \dfrac{sin (x)}{sin (x)}}[/tex]

[tex]\sf{a = \dfrac{cos (x) - 1 + sin (x)}{sin (x)}}[/tex]

Penyebut

[tex]\sf{b = cot (x) + csc (x) - 1}[/tex]

[tex]\sf{b = cot (x) + cosec (x) - 1}[/tex]

[tex]\sf{b = \dfrac{cos (x)}{sin (x)} + \dfrac{1}{sin (x)} - \dfrac{sin (x)}{sin (x)}}[/tex]

[tex]\sf{b = \dfrac{cos (x) + 1 - sin (x)}{sin (x)}}[/tex]

• Subtitusikan nilai a dan b

[tex]\sf{\dfrac{cot (x) - csc (x) + 1}{cot (x) + csc (x) - 1} = \dfrac{a}{b}}[/tex]

[tex]\sf{\dfrac{a}{b} = a \div b}[/tex]

[tex]\sf{\dfrac{a}{b} = \dfrac{cos (x) - 1 + sin (x)}{sin (x)} \div \dfrac{cos (x) + 1 - sin (x)}{sin (x)}}[/tex]

[tex]\sf{\dfrac{a}{b} = \dfrac{cos (x) - 1 + sin (x)}{sin (x)} \times \dfrac{sin (x)}{cos (x) + 1 - sin (x)}}[/tex]

[tex]\sf{\dfrac{a}{b} = \dfrac{cos (x) - 1 + sin (x)}{cos (x) + 1 - sin (x)}}[/tex]

• Maka diperoleh nilai a dan b

[tex]\sf{a = cos (x) - 1 + sin (x)}[/tex]

[tex]\sf{b = cos (x) + 1 - sin (x)}[/tex]

• Ubah nilai a dan b ke bentuk perkalian dengan menggunakan identitas trigonometri berikut

Sinus

[tex]\sf{sin (2a) = 2 sin (a) \: cos (a)}[/tex]

Jika [tex]\sf{a = \dfrac{x}{2}}[/tex] maka

[tex]\sf{sin (x) = 2 sin \left( \dfrac{x}{2} \right) \: cos \left( \dfrac{x}{2} \right)}[/tex]

Cosinus

[tex]\sf{cos (2a) = 2 {cos}^2 (a) - 1}[/tex]

[tex]\sf{cos (2a) = 1 - 2 {sin}^2 (a)}[/tex]

Jika [tex]\sf{a = \dfrac{x}{2}}[/tex] maka

[tex]\sf{cos (x) = 2 {cos}^2 \left( \dfrac{x}{2} \right) - 1}[/tex]

[tex]\sf{cos \: x = 1 - 2 {sin}^2 \left( \dfrac{x}{2} \right)}[/tex]

• Subtitusikan ke a

[tex]\sf{a = cos (x) - 1 + sin (x)}[/tex]

[tex]\sf{a = 1 - 2 {sin}^2 \left( \dfrac{x}{2} \right) - 1 + 2 sin \left( \dfrac{x}{2} \right) \: cos \left( \dfrac{x}{2} \right)}[/tex]

[tex]\sf{a = - 2 {sin}^2 \left( \dfrac{x}{2} \right) + 2 sin \left( \dfrac{x}{2} \right) \: cos \left( \dfrac{x}{2} \right)}[/tex]

[tex]\sf{a = sin \left( \dfrac{x}{2} \right) \: \left( -2 sin \left( \dfrac{x}{2} \right) + 2 cos \left( \dfrac{x}{2} \right) \right) }[/tex]

[tex]\sf{a = sin \left( \dfrac{x}{2} \right) \: \left( 2 cos \left( \dfrac{x}{2} \right) -2 sin \left( \dfrac{x}{2} \right) \right) }[/tex]

• Subtitusikan ke b

[tex]\sf{b = cos (x) + 1 - sin (x)}[/tex]

[tex]\sf{b = 2 {cos}^2 \left( \dfrac{x}{2} \right) - 1 + 1 - 2 sin \left( \dfrac{x}{2} \right) \: cos \left( \dfrac{x}{2} \right)}[/tex]

[tex]\sf{b = 2 {cos}^2 \left( \dfrac{x}{2} \right) - 2 sin \left( \dfrac{x}{2} \right) \: cos \left( \dfrac{x}{2} \right) }[/tex]

[tex]\sf{b = cos \left( \dfrac{x}{2} \right) \: \left( 2 cos \left( \dfrac{x}{2} \right) - 2 sin \left( \dfrac{x}{2} \right) \right) }[/tex]

• Subtitusikan nilai a dan b

[tex]\sf{\dfrac{cot (x) - csc (x) + 1}{cot (x) + csc (x) - 1}}[/tex]

[tex]\sf{= \dfrac{a}{b}}[/tex]

[tex]\sf{= \dfrac{sin \left( \dfrac{x}{2} \right) \: \left( 2 cos \left( \dfrac{x}{2} \right) -2 sin \left( \dfrac{x}{2} \right) \right) }{ cos \left( \dfrac{x}{2} \right) \left( 2 cos \left( \dfrac{x}{2} \right) - 2 sin \left( \dfrac{x}{2} \right) \right)}}[/tex]

[tex]\sf{= \dfrac{sin \left( \dfrac{x}{2} \right) }{ cos \left( \dfrac{x}{2} \right) }}[/tex]

[tex]\sf{= tan \left( \dfrac{x}{2} \right)}[/tex]

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Detail jawaban :

• Kelas : 10
• Mapel : Matematika
• Bab : 7 - Trigonometri
• Kode : 10.2.7
• Kata kunci : sin, cos, tan