Question 8
Difficulty: Ill
How many solutions, on the interval $] 0,2 \pi[$, are there for the equation, $|\tan \theta|=1$ ?
12
24
36
40

Answer

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Answer

$\boxed{4}$ solutions for the equation $|\tan \theta| = 1$ in the interval $(0, 2\pi)$

Steps

Step 1 ：Find the solutions for the equation $\tan \theta = 1$ and $\tan \theta = -1$ in the interval $(0, 2\pi)$

Step 2 ：$\tan \theta = 1$ when $\theta = \frac{\pi}{4} + n\pi$ and $\tan \theta = -1$ when $\theta = \frac{3\pi}{4} + n\pi$, where n is an integer

Step 3 ：Find the solutions that fall within the given interval:

Step 4 ：solutions_1 = $\left[\frac{\pi}{4}, \frac{5\pi}{4}\right]$

Step 5 ：solutions_2 = $\left[\frac{3\pi}{4}, \frac{7\pi}{4}\right]$

Step 6 ：total_solutions = 4

Step 7 ：$\boxed{4}$ solutions for the equation $|\tan \theta| = 1$ in the interval $(0, 2\pi)$