Ta có:

$sin~2x = cos\frac{\pi}{5}$

`\Leftrightarrow cos(\frac{\pi}{2}-2x) = cos\frac{\pi}{5}`

`\Leftrightarrow \frac{\pi}{2}-2x = \pm\frac{\pi}{5} + k2\pi, k \in \mathbb{Z}`

Trường hợp 1: `\frac{\pi}{2}-2x = \frac{\pi}{5} + k2\pi`

`\Leftrightarrow 2x = \frac{3\pi}{10} - k2\pi`

`\Leftrightarrow x = \frac{3\pi}{20} - k\pi`

Vì `0 < x < 4\pi` ta có:

`0 < \frac{3\pi}{20} - k\pi < 4\pi`

`\Leftrightarrow -\frac{3}{20} < -k < 4 - \frac{3}{20}`

`\Leftrightarrow \frac{3}{20} > k > -4 + \frac{3}{20}`

`\Leftrightarrow -3.85 < k < 0.15`

Vì `k` là số nguyên nên `k` có các giá trị là: `-3, -2, -1, 0`

`x_1 = \frac{3\pi}{20}`

`x_2 = \frac{3\pi}{20} + \pi = \frac{23\pi}{20}`

`x_3 = \frac{3\pi}{20} + 2\pi = \frac{43\pi}{20}`

`x_4 = \frac{3\pi}{20} + 3\pi = \frac{63\pi}{20}`

Trường hợp 2: `\frac{\pi}{2}-2x = -\frac{\pi}{5} + k2\pi`

`\Leftrightarrow 2x = \frac{7\pi}{10} - k2\pi`

`\Leftrightarrow x = \frac{7\pi}{20} - k\pi`

Vì `0 < x < 4\pi` ta có:

`0 < \frac{7\pi}{20} - k\pi < 4\pi`

`\Leftrightarrow -\frac{7}{20} < -k < 4 - \frac{7}{20}`

`\Leftrightarrow \frac{7}{20} > k > -4 + \frac{7}{20}`

`\Leftrightarrow -3.65 < k < 0.35`

Vì `k` là số nguyên nên `k` có các giá trị là: `-3, -2, -1, 0`

`x_5 = \frac{7\pi}{20}`

`x_6 = \frac{7\pi}{20} + \pi = \frac{27\pi}{20}`

`x_7 = \frac{7\pi}{20} + 2\pi = \frac{47\pi}{20}`

`x_8 = \frac{7\pi}{20} + 3\pi = \frac{67\pi}{20}`

Tổng các nghiệm trong khoảng `(0;4\pi)` là:

`S = \frac{3\pi}{20} + \frac{23\pi}{20} + \frac{43\pi}{20} + \frac{63\pi}{20} + \frac{7\pi}{20} + \frac{27\pi}{20} + \frac{47\pi}{20} + \frac{67\pi}{20}`

`S = \frac{320\pi}{20} = 14\pi`

`-> bbC`