To solve this problem, follow these step-by-step instructions:
1. Determine [tex]\(\sin \alpha\)[/tex]:
Given the equation [tex]\(3 \sin \alpha = 1\)[/tex], we can solve for [tex]\(\sin \alpha\)[/tex] as follows:
[tex]\[
\sin \alpha = \frac{1}{3}
\][/tex]
2. Find [tex]\(\cos \alpha\)[/tex] using the Pythagorean identity:
The Pythagorean identity states that:
[tex]\[
\sin^2 \alpha + \cos^2 \alpha = 1
\][/tex]
Substitute [tex]\(\sin \alpha = \frac{1}{3}\)[/tex]:
[tex]\[
\left(\frac{1}{3}\right)^2 + \cos^2 \alpha = 1
\][/tex]
Simplify the equation:
[tex]\[
\frac{1}{9} + \cos^2 \alpha = 1
\][/tex]
[tex]\[
\cos^2 \alpha = 1 - \frac{1}{9}
\][/tex]
[tex]\[
\cos^2 \alpha = \frac{8}{9}
\][/tex]
Thus, [tex]\(\cos \alpha\)[/tex] is:
[tex]\[
\cos \alpha = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}
\][/tex]
3. Calculate [tex]\(\tan \alpha\)[/tex]:
The tangent function is defined as:
[tex]\[
\tan \alpha = \frac{\sin \alpha}{\cos \alpha}
\][/tex]
Using [tex]\(\sin \alpha = \frac{1}{3}\)[/tex] and [tex]\(\cos \alpha = \frac{2\sqrt{2}}{3}\)[/tex]:
[tex]\[
\tan \alpha = \frac{\frac{1}{3}}{\frac{2\sqrt{2}}{3}} = \frac{1}{2\sqrt{2}} = \frac{1}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{4}
\][/tex]
4. Compute the value of the given expression:
We need to find:
[tex]\[
\frac{2 \tan \alpha}{1 - \tan^2 \alpha}
\][/tex]
Substitute [tex]\(\tan \alpha = \frac{\sqrt{2}}{4}\)[/tex]:
[tex]\[
1 - \tan^2 \alpha = 1 - \left( \frac{\sqrt{2}}{4} \right)^2 = 1 - \frac{2}{16} = 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8}
\][/tex]
Now, calculate the expression:
[tex]\[
\frac{2 \cdot \frac{\sqrt{2}}{4}}{1 - \left( \frac{\sqrt{2}}{4} \right)^2} = \frac{ \frac{2\sqrt{2}}{4}}{\frac{7}{8}} = \frac{\sqrt{2}}{2} \cdot \frac{8}{7} = \frac{\sqrt{2} \cdot 8}{2 \cdot 7} = \frac{8\sqrt{2}}{14} = \frac{4\sqrt{2}}{7}
\][/tex]
Therefore, the value of the expression [tex]\(\frac{2 \tan \alpha}{1 - \tan^2 \alpha}\)[/tex] is [tex]\(\boxed{0.8081220356417685}\)[/tex].
