Trigonometri Contoh

\tan (2 x) = \frac{2}{\cot (x) - \tan (x)}

$\tan (2 x) = \frac{2}{\cot (x) - \tan (x)}$

Langkah 1

Mulai dari sisi kiri.

\tan (2 x)

$\tan (2 x)$

Langkah 2

Terapkan identitas sudut ganda tangen.

\frac{2 \tan (x)}{1 - \tan^{2} (x)}

$\frac{2 \tan (x)}{1 - \tan^{2} (x)}$

Langkah 3

Konversikan ke sinus dan kosinus.

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Langkah 3.1

Tulis

\tan (x)

$\tan (x)$ dalam sinus dan kosinus menggunakan identitas hasil bagi.

\frac{2 \frac{\sin (x)}{\cos (x)}}{1 - \tan^{2} (x)}

$\frac{2 \frac{\sin (x)}{\cos (x)}}{1 - \tan^{2} (x)}$

Langkah 3.2

Tulis

\tan (x)

$\tan (x)$ dalam sinus dan kosinus menggunakan identitas hasil bagi.

\frac{2 \frac{\sin (x)}{\cos (x)}}{1 - {(\frac{\sin (x)}{\cos (x)})}^{2}}

$\frac{2 \frac{\sin (x)}{\cos (x)}}{1 - {(\frac{\sin (x)}{\cos (x)})}^{2}}$

Langkah 3.3

Terapkan kaidah hasil kali ke

\frac{\sin (x)}{\cos (x)}

$\frac{\sin (x)}{\cos (x)}$ .

\frac{2 \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin^{2} (x)}{\cos^{2} (x)}}

$\frac{2 \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin^{2} (x)}{\cos^{2} (x)}}$

\frac{2 \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin^{2} (x)}{\cos^{2} (x)}}

$\frac{2 \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin^{2} (x)}{\cos^{2} (x)}}$

Langkah 4

Sederhanakan.

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Langkah 4.1

Gabungkan

2

$2$ dan

\frac{\sin (x)}{\cos (x)}

$\frac{\sin (x)}{\cos (x)}$ .

\frac{\frac{2 \sin (x)}{\cos (x)}}{1 - \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{1 - \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}}$

Langkah 4.2

Sederhanakan penyebutnya.

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Langkah 4.2.1

Tulis kembali

1

$1$ sebagai

1^{2}

$1^{2}$ .

\frac{\frac{2 \sin (x)}{\cos (x)}}{1^{2} - \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{1^{2} - \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}}$

Langkah 4.2.2

Tulis kembali

\frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}

$\frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ sebagai

{(\frac{\sin (x)}{\cos (x)})}^{2}

${(\frac{\sin (x)}{\cos (x)})}^{2}$ .

\frac{\frac{2 \sin (x)}{\cos (x)}}{1^{2} - {(\frac{\sin (x)}{\cos (x)})}^{2}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{1^{2} - {(\frac{\sin (x)}{\cos (x)})}^{2}}$

Langkah 4.2.3

Karena kedua suku merupakan kuadrat sempurna, faktorkan menggunakan rumus beda pangkat dua,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ di mana

a = 1

$a = 1$ dan

b = \frac{\sin (x)}{\cos (x)}

$b = \frac{\sin (x)}{\cos (x)}$ .

\frac{\frac{2 \sin (x)}{\cos (x)}}{(1 + \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{(1 + \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})}$

Langkah 4.2.4

Tuliskan

1

$1$ sebagai pecahan dengan penyebut persekutuan.

\frac{\frac{2 \sin (x)}{\cos (x)}}{(\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{(\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})}$

Langkah 4.2.5

Gabungkan pembilang dari penyebut persekutuan.

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{\cos (x) + \sin (x)}{\cos (x)} (1 - \frac{\sin (x)}{\cos (x)})}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{\cos (x) + \sin (x)}{\cos (x)} (1 - \frac{\sin (x)}{\cos (x)})}$

Langkah 4.2.6

Tuliskan

1

$1$ sebagai pecahan dengan penyebut persekutuan.

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{\cos (x) + \sin (x)}{\cos (x)} (\frac{\cos (x)}{\cos (x)} - \frac{\sin (x)}{\cos (x)})}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{\cos (x) + \sin (x)}{\cos (x)} (\frac{\cos (x)}{\cos (x)} - \frac{\sin (x)}{\cos (x)})}$

Langkah 4.2.7

Gabungkan pembilang dari penyebut persekutuan.

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{\cos (x) + \sin (x)}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x)}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{\cos (x) + \sin (x)}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x)}}$

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{\cos (x) + \sin (x)}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x)}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{\cos (x) + \sin (x)}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x)}}$

Langkah 4.3

Kalikan

\frac{\cos (x) + \sin (x)}{\cos (x)}

$\frac{\cos (x) + \sin (x)}{\cos (x)}$ dengan

\frac{\cos (x) - \sin (x)}{\cos (x)}

$\frac{\cos (x) - \sin (x)}{\cos (x)}$ .

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{\cos (x) \cos (x)}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{\cos (x) \cos (x)}}$

Langkah 4.4

Sederhanakan penyebutnya.

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Langkah 4.4.1

Naikkan

\cos (x)

$\cos (x)$ menjadi pangkat

1

$1$ .

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{1} \cos (x)}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{1} \cos (x)}}$

Langkah 4.4.2

Naikkan

\cos (x)

$\cos (x)$ menjadi pangkat

1

$1$ .

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{1} {\cos (x)}^{1}}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{1} {\cos (x)}^{1}}}$

Langkah 4.4.3

Gunakan kaidah pangkat

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{1 + 1}}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{1 + 1}}}$

Langkah 4.4.4

Tambahkan

1

$1$ dan

1

$1$ .

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{2}}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{2}}}$

\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{2}}}

$\frac{\frac{2 \sin (x)}{\cos (x)}}{\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\cos (x)}^{2}}}$

Langkah 4.5

Kalikan pembilang dengan balikan dari penyebut.

\frac{2 \sin (x)}{\cos (x)} \cdot \frac{{\cos (x)}^{2}}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}

$\frac{2 \sin (x)}{\cos (x)} \cdot \frac{{\cos (x)}^{2}}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}$

Langkah 4.6

Gabungkan.

\frac{2 \sin (x) {\cos (x)}^{2}}{\cos (x) ((\cos (x) + \sin (x)) (\cos (x) - \sin (x)))}

$\frac{2 \sin (x) {\cos (x)}^{2}}{\cos (x) ((\cos (x) + \sin (x)) (\cos (x) - \sin (x)))}$

Langkah 4.7

Hapus faktor persekutuan dari

{\cos (x)}^{2}

${\cos (x)}^{2}$ dan

\cos (x)

$\cos (x)$ .

\frac{2 \sin (x) \cos (x)}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}

$\frac{2 \sin (x) \cos (x)}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}$

\frac{2 \sin (x) \cos (x)}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}

$\frac{2 \sin (x) \cos (x)}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}$

Langkah 5

Susun kembali suku-suku.

\frac{2 \cos (x) \sin (x)}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}

$\frac{2 \cos (x) \sin (x)}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}$

Langkah 6

Tulis kembali

\frac{2 \cos (x) \sin (x)}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}

$\frac{2 \cos (x) \sin (x)}{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}$ sebagai

\frac{2}{\cot (x) - \tan (x)}

$\frac{2}{\cot (x) - \tan (x)}$ .

\frac{2}{\cot (x) - \tan (x)}

$\frac{2}{\cot (x) - \tan (x)}$

Langkah 7

Karena kedua sisi telah terbukti setara, maka persamaan tersebut adalah sebuah identitas.

\tan (2 x) = \frac{2}{\cot (x) - \tan (x)}

$\tan (2 x) = \frac{2}{\cot (x) - \tan (x)}$ adalah identitas