To determine which of the given options are solutions to the equation [tex]\(\tan^2 x - 3 = 0\)[/tex], let's proceed step by step.
### Step 1: Solve the equation symbolically
First, we need to solve the given equation:
[tex]\[
\tan^2(x) - 3 = 0
\][/tex]
Adding 3 to both sides, we get:
[tex]\[
\tan^2(x) = 3
\][/tex]
Taking the square root of both sides, we obtain:
[tex]\[
\tan(x) = \pm \sqrt{3}
\][/tex]
This gives us two possibilities to consider:
1. [tex]\(\tan(x) = \sqrt{3}\)[/tex]
2. [tex]\(\tan(x) = -\sqrt{3}\)[/tex]
### Step 2: Identify the angles that satisfy the conditions
For [tex]\(\tan(x) = \sqrt{3}\)[/tex], the primary solutions within the range [tex]\([0, 2\pi)\)[/tex] are:
- [tex]\(x = \frac{\pi}{3}\)[/tex], and any angle coterminal with it, i.e., [tex]\(x = \frac{\pi}{3} + k\pi\)[/tex] for any integer [tex]\(k\)[/tex]
For [tex]\(\tan(x) = -\sqrt{3}\)[/tex], the primary solutions within the range [tex]\([0, 2\pi)\)[/tex] are:
- [tex]\(x = \frac{2\pi}{3}\)[/tex], and any angle coterminal with it, i.e., [tex]\(x = \frac{2\pi}{3} + k\pi\)[/tex] for any integer [tex]\(k\)[/tex]
### Step 3: Evaluate the given options
Let's now evaluate the given options to see if they match any of our derived solutions:
- Option A: [tex]\(x = -\frac{\pi}{3}\)[/tex]
[tex]\[
\tan\left(-\frac{\pi}{3}\right) = - \sqrt{3}
\][/tex]
The equation requires [tex]\(\tan^2(x) = 3\)[/tex], so:
[tex]\[
(-\sqrt{3})^2 = 3
\][/tex]
Substituting back, [tex]\(\tan^2\left(-\frac{\pi}{3}\right) - 3= 0\)[/tex], we need:
[tex]\[
3 - 3 = 0
\][/tex]
This yields true.
- Option B: [tex]\(x = \frac{5\pi}{3}\)[/tex]
[tex]\[
\tan\left(\frac{5\pi}{3}\right) = - \sqrt{3}
\][/tex]
The equation requires [tex]\(\tan^2(x) = 3\)[/tex], so:
[tex]\[
(-\sqrt{3})^2 = 3
\][/tex]
Substituting back, [tex]\(\tan^2\left(\frac{5\pi}{3}\right) - 3= 0\)[/tex], we need:
[tex]\[
3 - 3 = 0
\][/tex]
This yields true.
- Option C: [tex]\(x = \pi\)[/tex]
[tex]\[
\tan(\pi) = 0
\][/tex]
The equation requires [tex]\(\tan^2(x) = 3\)[/tex], so:
[tex]\[
0^2 = 0\neq 3
\][/tex]
Substituting back, [tex]\(\tan^2(\pi) - 3 \)[/tex], yields false.
- Option D: [tex]\(x = \frac{2\pi}{3}\)[/tex]
[tex]\[
\tan\left(\frac{2\pi}{3}\right) = - \sqrt{3}
\][/tex]
The equation requires [tex]\(\tan^2(x) = 3\)[/tex], so:
[tex]\[
(-\sqrt{3})^2 = 3
\][/tex]
Substituting back, [tex]\(\tan^2\left(\frac{2\pi}{3}\right) - 3= 0\)[/tex], we need:
[tex]\[
3 - 3 = 0
\][/tex]
This yields true.
### Conclusion:
From evaluating the options:
- Option A: [tex]\( -\frac{\pi}{3} \)[/tex]
- Option B: [tex]\( \frac{5\pi}{3} \)[/tex]
- Option D: [tex]\( \frac{2\pi}{3} \)[/tex]
These satisfy the given equation [tex]\(\tan^2 x - 3 = 0\)[/tex], whereas Option C: [tex]\(\pi\)[/tex] does not.
Thus, the correct solutions are:
A, B, and D.
