Trigonometri Contoh

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

Langkah 1

Sederhanakan setiap suku.

Ketuk untuk lebih banyak langkah...

Langkah 1.1

Tulis kembali $\tan (x)$ dalam bentuk sinus dan kosinus.

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

Langkah 1.2

Tulis kembali $\cot (x)$ dalam bentuk sinus dan kosinus.

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

Langkah 2

Untuk menuliskan $\frac{\cos (x)}{1 - \frac{\sin (x)}{\cos (x)}}$ sebagai pecahan dengan penyebut umum, kalikan dengan $\frac{1 - \frac{\cos (x)}{\sin (x)}}{1 - \frac{\cos (x)}{\sin (x)}}$ .

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

Langkah 3

Untuk menuliskan $\frac{\sin (x)}{1 - \frac{\cos (x)}{\sin (x)}}$ sebagai pecahan dengan penyebut umum, kalikan dengan $\frac{1 - \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin (x)}{\cos (x)}}$ .

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

Langkah 4

Tulis setiap pernyataan menggunakan penyebut umum dari $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ , dengan mengalikan masing-masing pembilang dan penyebut dengan faktor dari $1$ yang sesuai.

Ketuk untuk lebih banyak langkah...

Langkah 4.1

Kalikan $\frac{\cos (x)}{1 - \frac{\sin (x)}{\cos (x)}}$ dengan $\frac{1 - \frac{\cos (x)}{\sin (x)}}{1 - \frac{\cos (x)}{\sin (x)}}$ .

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

Langkah 4.2

Kalikan $\frac{\sin (x)}{1 - \frac{\cos (x)}{\sin (x)}}$ dengan $\frac{1 - \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin (x)}{\cos (x)}}$ .

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

Langkah 4.3

Susun kembali faktor-faktor dari $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ .

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

Langkah 5

Gabungkan pembilang dari penyebut persekutuan.

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

Langkah 6

Sederhanakan pembilangnya.

Ketuk untuk lebih banyak langkah...

Langkah 6.1

Terapkan sifat distributif.

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

Langkah 6.2

Kalikan $\cos (x)$ dengan $1$ .

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

Langkah 6.3

Tulis kembali menggunakan sifat komutatif dari perkalian.

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

Langkah 6.4

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

Ketuk untuk lebih banyak langkah...

Langkah 6.4.1

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

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

Langkah 6.4.2

Naikkan $\cos (x)$ menjadi pangkat $1$ .

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

Langkah 6.4.3

Naikkan $\cos (x)$ menjadi pangkat $1$ .

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

Langkah 6.4.4

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

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

Langkah 6.4.5

Tambahkan $1$ dan $1$ .

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

Langkah 6.5

Terapkan sifat distributif.

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

Langkah 6.6

Kalikan $\sin (x)$ dengan $1$ .

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

Langkah 6.7

Tulis kembali menggunakan sifat komutatif dari perkalian.

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

Langkah 6.8

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

Ketuk untuk lebih banyak langkah...

Langkah 6.8.1

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

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

Langkah 6.8.2

Naikkan $\sin (x)$ menjadi pangkat $1$ .

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

Langkah 6.8.3

Naikkan $\sin (x)$ menjadi pangkat $1$ .

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

Langkah 6.8.4

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

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

Langkah 6.8.5

Tambahkan $1$ dan $1$ .

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

Langkah 7

Sederhanakan pembilangnya.

Ketuk untuk lebih banyak langkah...

Langkah 7.1

Faktorkan $\cos (x)$ dari $\cos^{2} (x)$ .

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

Langkah 7.2

Pisahkan pecahan.

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

Langkah 7.3

Konversikan dari $\frac{\cos (x)}{\sin (x)}$ ke $\cot (x)$ .

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

Langkah 7.4

Bagilah $\cos (x)$ dengan $1$ .

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

Langkah 7.5

Faktorkan $\sin (x)$ dari $\sin^{2} (x)$ .

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

Langkah 7.6

Pisahkan pecahan.

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

Langkah 7.7

Konversikan dari $\frac{\sin (x)}{\cos (x)}$ ke $\tan (x)$ .

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

Langkah 7.8

Bagilah $\sin (x)$ dengan $1$ .

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

Langkah 8

Sederhanakan penyebutnya.

Ketuk untuk lebih banyak langkah...

Langkah 8.1

Konversikan dari $\frac{\cos (x)}{\sin (x)}$ ke $\cot (x)$ .

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

Langkah 8.2

Konversikan dari $\frac{\sin (x)}{\cos (x)}$ ke $\tan (x)$ .

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