To solve this problem, let's analyze the equation and evaluate which angles do not satisfy it.

The given equation is: $(3 \tan^2 x - 1)(3 \tan^2 x - 3) = 0$

This equation will hold true if either factor equals zero. So, let's split it into two cases:

Case 1: $3 \tan^2 x - 1 = 0$

$3 \tan^2 x = 1 \Rightarrow \tan^2 x = \frac{1}{3} \Rightarrow \tan x = \pm \frac{1}{\sqrt{3}}$

The angles that satisfy $\tan x = \frac{1}{\sqrt{3}}$ or $\tan x = -\frac{1}{\sqrt{3}}$ in the interval $[0^\circ, 360^\circ]$ are:

• $\tan x = \frac{1}{\sqrt{3}} \Rightarrow x = 30^\circ, 210^\circ$
• $\tan x = -\frac{1}{\sqrt{3}} \Rightarrow x = 150^\circ, 330^\circ$

Case 2: $3 \tan^2 x - 3 = 0$

$3 \tan^2 x = 3 \Rightarrow \tan^2 x = 1 \Rightarrow \tan x = \pm 1$

The angles that satisfy $\tan x = 1$ or $\tan x = -1$ are:

• $\tan x = 1 \Rightarrow x = 45^\circ, 225^\circ$
• $\tan x = -1 \Rightarrow x = 135^\circ, 315^\circ$

Conclusion

Now, let's list all the angles that satisfy either part of the equation:

• From Case 1: $30^\circ, 150^\circ, 210^\circ, 330^\circ$
• From Case 2: $45^\circ, 135^\circ, 225^\circ, 315^\circ$

Among the options provided, we need to find the angle that does not appear in either list:

• $30^\circ$ is in Case 1.
• $120^\circ$ does not appear in either list.
• $150^\circ$ is in Case 1.
• $225^\circ$ is in Case 2.
• $315^\circ$ is in Case 2.

Answer

The correct answer is: B. $120^\circ$

Would you like further clarification on any step?

Here are some additional questions that could deepen your understanding of this topic:

1. How would the solution change if the equation had a different trigonometric function, like sine or cosine?
2. What are the solutions if the equation included a phase shift or vertical shift in the tangent function?
3. Could this type of factorization approach be used with equations involving other trigonometric identities?
4. How would the solution be affected if we changed the range of angles to radians or used a different interval?
5. Can you determine which other angles in different quadrants satisfy the same tangent conditions?

Tip: When solving trigonometric equations, remember to consider all possible angles within the given interval that satisfy each solution!