Question

How many solutions does the equation sin(x) sin(2x) sin(3x)...sin(11x) sin(12x) = 0
have in the interval (0,π ] ?

Answer 1

Answer on Question #72205 – Math – Trigonometry

Question

How many solutions does the equation

$$\sin(x) \sin(2x) \sin(3x) \dots \sin(11x) \sin(12x) = 0$$

have in the interval $(0, \pi]$ ?

Solution

The equation $\sin(12x) = 0$ has solutions in the interval $(0, \pi]$

$$x_1 = \frac{\pi}{12}, \quad x_2 = \frac{\pi}{6}, \quad x_3 = \frac{\pi}{4}, \quad x_4 = \frac{\pi}{3}, \quad x_5 = \frac{5\pi}{12}, \quad x_6 = \frac{\pi}{2}, \quad x_7 = \frac{7\pi}{12}, \quad x_8 = \frac{2\pi}{3},$$ $$x_9 = \frac{3\pi}{4}, \quad x_{10} = \frac{5\pi}{6}, \quad x_{11} = \frac{11\pi}{12}, \quad x_{12} = \pi$$

The equation $\sin(11x) = 0$ has solutions in the interval $(0, \pi]$

$$x_1 = \frac{\pi}{11}, \quad x_2 = \frac{2\pi}{11}, \quad x_3 = \frac{3\pi}{11}, \quad x_4 = \frac{4\pi}{11}, \quad x_5 = \frac{5\pi}{11}, \quad x_6 = \frac{6\pi}{11}, \quad x_7 = \frac{7\pi}{11}, \quad x_8 = \frac{8\pi}{11},$$ $$x_9 = \frac{9\pi}{11}, \quad x_{10} = \frac{10\pi}{11}, \quad x_{11} = \pi$$

The equation $\sin(10x) = 0$ has solutions in the interval $(0, \pi]$

$$x_1 = \frac{\pi}{10}, \quad x_2 = \frac{\pi}{5}, \quad x_3 = \frac{3\pi}{10}, \quad x_4 = \frac{2\pi}{5}, \quad x_5 = \frac{\pi}{2}, \quad x_6 = \frac{3\pi}{5}, \quad x_7 = \frac{7\pi}{10}, \quad x_8 = \frac{4\pi}{5},$$ $$x_9 = \frac{9\pi}{10}, \quad x_{10} = \pi$$

The equation $\sin(9x) = 0$ has solutions in the interval $(0, \pi]$

$$x_1 = \frac{\pi}{9}, \quad x_2 = \frac{2\pi}{9}, \quad x_3 = \frac{\pi}{3}, \quad x_4 = \frac{4\pi}{9}, \quad x_5 = \frac{5\pi}{9}, \quad x_6 = \frac{2\pi}{3}, \quad x_7 = \frac{7\pi}{9}, \quad x_8 = \frac{8\pi}{9},$$ $$x_9 = \pi$$

The equation $\sin(8x) = 0$ has solutions in the interval $(0, \pi]$

$$x_1 = \frac{\pi}{8}, \quad x_2 = \frac{\pi}{4}, \quad x_3 = \frac{3\pi}{8}, \quad x_4 = \frac{\pi}{2}, \quad x_5 = \frac{5\pi}{8}, \quad x_6 = \frac{3\pi}{4}, \quad x_7 = \frac{7\pi}{8}, \quad x_8 = \pi$$

The equation $\sin(7x) = 0$ has solutions in the interval $(0, \pi]$

$$x_1 = \frac{\pi}{7}, \quad x_2 = \frac{2\pi}{7}, \quad x_3 = \frac{3\pi}{7}, \quad x_4 = \frac{4\pi}{7}, \quad x_5 = \frac{5\pi}{7}, \quad x_6 = \frac{6\pi}{7}, \quad x_7 = \pi$$

Therefore, the equation $\sin(x)\sin(2x)\sin(3x)\dots\sin(11x)\sin(12x) = 0$ has solutions in the interval $(0,\pi]$

$$x _ {1} = \frac {\pi}{1 2}, x _ {2} = \frac {\pi}{1 1}, x _ {3} = \frac {\pi}{1 0}, x _ {4} = \frac {\pi}{9}, x _ {5} = \frac {\pi}{8}, x _ {6} = \frac {\pi}{7}, x _ {7} = \frac {\pi}{6}, x _ {8} = \frac {2 \pi}{1 1},$$ $$x _ {9} = \frac {\pi}{5}, x _ {1 0} = \frac {2 \pi}{9}, x _ {1 1} = \frac {\pi}{4}, x _ {1 2} = \frac {2 \pi}{7}, x _ {1 3} = \frac {3 \pi}{1 1}, x _ {1 4} = \frac {3 \pi}{1 0}, x _ {1 5} = \frac {\pi}{3}, x _ {1 6} = \frac {3 \pi}{8},$$ $$x _ {1 7} = \frac {3 \pi}{7}, x _ {1 8} = \frac {4 \pi}{1 1}, x _ {1 9} = \frac {2 \pi}{5}, x _ {2 0} = \frac {4 \pi}{9}, x _ {2 1} = \frac {\pi}{2}, x _ {2 2} = \frac {4 \pi}{7}, x _ {2 3} = \frac {5 \pi}{1 2},$$ $$x _ {2 4} = \frac {5 \pi}{1 1}, x _ {2 5} = \frac {5 \pi}{9}, x _ {2 6} = \frac {5 \pi}{8}, x _ {2 7} = \frac {5 \pi}{7}, x _ {2 8} = \frac {6 \pi}{1 1}, x _ {2 9} = \frac {3 \pi}{5}, x _ {3 0} = \frac {2 \pi}{3},$$ $$x _ {3 1} = \frac {3 \pi}{4}, x _ {3 2} = \frac {6 \pi}{7}, x _ {3 3} = \frac {7 \pi}{1 2}, x _ {3 4} = \frac {7 \pi}{1 1}, x _ {3 5} = \frac {7 \pi}{1 0}, x _ {3 6} = \frac {7 \pi}{9}, x _ {3 7} = \frac {7 \pi}{8},$$ $$x _ {3 8} = \frac {8 \pi}{1 1}, x _ {3 9} = \frac {4 \pi}{5}, x _ {4 0} = \frac {8 \pi}{9}, x _ {4 1} = \frac {9 \pi}{1 1}, x _ {4 2} = \frac {9 \pi}{1 0}, x _ {4 3} = \frac {5 \pi}{6}, x _ {4 4} = \frac {1 0 \pi}{1 1},$$ $$x _ {4 5} = \frac {1 1 \pi}{1 2}, x _ {4 6} = \pi .$$

There are 46 solutions.

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