# Thread: Conditional Trigonometric Equations, Helpo!!

1. ## Conditional Trigonometric Equations, Helpo!!

Solve the equation for exact solutions over the interval 0,2

tan(squared)x + 3 = 0

2. Originally Posted by nee
Solve the equation for exact solutions over the interval 0,2

tan(squared)x + 3 = 0
$\displaystyle \tan^2x+3=0$

There is no such x value that causes this statement to be true.

Maybe you mean $\displaystyle \tan^2x{\color{red}-}3=0$??

--Chris

3. Originally Posted by nee
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.

4. That's the question as it's written in the text, and the answer given is 0 with a diagonal line through it, whatever that means.

5. Originally Posted by nee
That's the question as it's written in the text, and the answer given is 0 with a diagonal line through it, whatever that means.
LoL

$\displaystyle \emptyset$. This means that there is no solution. More precisely $\displaystyle \{\emptyset\}$ stands for "the set of the solutions is empty"

6. Oh ok. So what kind of work I'm supposed to show to demonstrate how I came to that conclusion?

7. Originally Posted by nee
Oh ok. So what kind of work I'm supposed to show to demonstrate how I came to that conclusion?
Note that you can't have $\displaystyle \tan\vartheta=\pm\sqrt{3}i$

Since the tangent would be complex, there wouldn't be a possible solution for x. So our answer would just be an empty set, $\displaystyle \left\{\emptyset\right\}$

8. Why can't you have ? When I punch it into my calculator I get a number.

9. Originally Posted by nee
Why can't you have ? When I punch it into my calculator I get a number.
An imaginary number isn't on the x-axis. In order for that number to be a solution to the equation, the equation must intersect the x-axis at that point. Since that point doesn't exist on the x-axis when y = 0, then it is not a solution of the equation.

10. Originally Posted by nee
Why can't you have ? When I punch it into my calculator I get a number.
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

11. Thanks!

12. Thanks!

nee

13. Hi,
Originally Posted by Moo
More precisely $\displaystyle \{\emptyset\}$ stands for "the set of the solutions is empty"
Originally Posted by Chris L T521
So our answer would just be an empty set, $\displaystyle \left\{\emptyset\right\}$
$\displaystyle \{\emptyset\}$ 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 $\displaystyle \emptyset$.