15. Find all solutions (in radians) for the equation \( \tan \left(\frac{\theta}{3}\right)+\sqrt{3}=0 \) (where \( k \), below, is any integer) A) \( -\frac{\pi}{2}+3 k \pi \) B) \( \frac{3 \pi}{4}+3 k \pi \) C) \( -\frac{3 \pi}{4}+3 k \pi \) D) \( -\pi+3 k \pi \) E) \( \pi+3 k \pi \)

To find the solutions to the equation \( \tan \left(\frac{\theta}{3}\right) + \sqrt{3} = 0 \), we first rewrite it as \( \tan \left(\frac{\theta}{3}\right) = -\sqrt{3} \). The tangent function is negative in the second and fourth quadrants, which correspond to angles of the form \( \frac{2\pi}{3} + k\pi \) for integer \( k \). Now we can set up the equation: \[ \frac{\theta}{3} = \frac{2\pi}{3} + k\pi \] Multiplying through by 3 gives: \[ \theta = 2\pi + 3k\pi \] Thus, we can express the general solution as: \[ \theta = 2\pi + 3k\pi \] From the reference angles, tangent gives us certain standard angles, so we can also check the equivalent form for \( \frac{\theta}{3} = \frac{5\pi}{3} + k\pi \) which results in \( \theta = 5\pi + 3k\pi \). Let's rewrite those in the form of \( \theta \): - Option A: \( -\frac{\pi}{2}+3k\pi \) does not work. - Option B: \( \frac{3 \pi}{4}+3k\pi \) does not work. - Option C: \( -\frac{3 \pi}{4}+3k\pi \) does not work. - Option D: \( -\pi+3k\pi \) does not work. - Option E: \( \pi+3k\pi \) does not work. In conclusion, we can simplify our options further. Based on our analysis, the correct answer for solutions is: \[ \theta = 2\pi + 3k\pi \quad \text{and} \quad \theta = 5\pi + 3k\pi \] All should be multiples of \( \frac{\pi}{3}\) corresponding to these transformations.