Solve the equation for exact solutions over the interval 0,2∏
tan(squared)x + 3 = 0
The one you gave doesn't exist so I'm assuming you made a typo.
Add 3, then take the square root of both sides. Look on the unit circle to see which angles correspond to $\displaystyle tan = \sqrt{3}$ over the interval $\displaystyle [0,2\pi)$, which is pretty much the entire unit circle. Those points will be your solutions.
Maybe I should have said this: You will not find an appropriate value for $\displaystyle \vartheta$ that falls in the domain $\displaystyle 0\leq\vartheta\leq 2\pi$. By the way, my calculator tells me that $\displaystyle \vartheta=\pm\frac{\pi}{2}\mp\frac{\ln(2-\sqrt{3})}{2}i$ which apparently seems like it won't fall in the interval $\displaystyle 0\leq\vartheta\leq 2\pi$.
--Chris
Hi,
$\displaystyle \{\emptyset\}$ is the set whose only member is the set $\displaystyle \emptyset$. Saying that the set of the solutions of $\displaystyle \tan^2x+3=0$ is $\displaystyle \{\emptyset\}$ means that for $\displaystyle x = \emptyset$ the equation is satisfied. In other words, $\displaystyle \tan^2\emptyset +3=0$ . I guess we agree that this doesn't make sense at all. To say that the set of the solutions is empty, simply get rid of the two braces : the set of the solutions of $\displaystyle \tan^2x+3=0$ is $\displaystyle \emptyset$.