Certainly! Let's solve the equation [tex]\(3 \tan \theta - \sqrt{3} = 0\)[/tex] in a detailed step-by-step manner.
### Step 1: Isolate the [tex]\(\tan \theta\)[/tex] term
First, add [tex]\(\sqrt{3}\)[/tex] to both sides of the equation to isolate the [tex]\(\tan \theta\)[/tex] term on the left-hand side:
[tex]\[
3 \tan \theta - \sqrt{3} + \sqrt{3} = 0 + \sqrt{3}
\][/tex]
This simplifies to:
[tex]\[
3 \tan \theta = \sqrt{3}
\][/tex]
### Step 2: Solve for [tex]\(\tan \theta\)[/tex]
Next, divide both sides of the equation by 3 to solve for [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sqrt{3}}{3}
\][/tex]
### Step 3: Find the angle [tex]\(\theta\)[/tex]
To find the angle [tex]\(\theta\)[/tex] that satisfies [tex]\(\tan \theta = \frac{\sqrt{3}}{3}\)[/tex], we use the inverse tangent function (also known as arctangent):
[tex]\[
\theta = \arctan\left(\frac{\sqrt{3}}{3}\right)
\][/tex]
### Step 4: Express [tex]\(\theta\)[/tex] in radians and degrees
The result of [tex]\(\arctan\left(\frac{\sqrt{3}}{3}\right)\)[/tex] is approximately [tex]\(0.5236\)[/tex] radians. To convert this to degrees, we use the conversion factor [tex]\(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)[/tex]:
[tex]\[
\theta \approx 0.5236 \text{ radians}
\][/tex]
To convert [tex]\(0.5236\)[/tex] radians to degrees:
[tex]\[
\theta \approx 0.5236 \times \left(\frac{180}{\pi}\right) \approx 30 \text{ degrees}
\][/tex]
### Step 5: Final Answer
Thus, the angle [tex]\(\theta\)[/tex] that satisfies the equation [tex]\(3 \tan \theta - \sqrt{3} = 0\)[/tex] is approximately:
[tex]\[
\theta \approx 0.5236 \text{ radians} \quad \text{or} \quad \theta \approx 30 \text{ degrees}
\][/tex]
Hence, these are the solutions for [tex]\(\theta\)[/tex].
