To find the value of [tex]\(\tan \theta\)[/tex] given that [tex]\(\sin \theta = \frac{2 \sqrt{x}}{x + 1}\)[/tex], we can follow these steps:
1. Compute [tex]\(\sin \theta\)[/tex]:
We are given:
[tex]\[
\sin \theta = \frac{2 \sqrt{x}}{x + 1}
\][/tex]
2. Calculate [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\sin^2 \theta = \left(\frac{2 \sqrt{x}}{x + 1}\right)^2 = \frac{4x}{(x + 1)^2}
\][/tex]
3. Use the Pythagorean identity to find [tex]\(\cos^2 \theta\)[/tex]:
We know that:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Therefore:
[tex]\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{4x}{(x + 1)^2}
\][/tex]
4. Simplify [tex]\(\cos^2 \theta\)[/tex]:
Finding a common denominator:
[tex]\[
\cos^2 \theta = \frac{(x + 1)^2 - 4x}{(x + 1)^2}
\][/tex]
Simplify the numerator:
[tex]\[
(x + 1)^2 - 4x = x^2 + 2x + 1 - 4x = x^2 - 2x + 1 = (x - 1)^2
\][/tex]
Therefore:
[tex]\[
\cos^2 \theta = \frac{(x - 1)^2}{(x + 1)^2}
\][/tex]
5. Find [tex]\(\cos \theta\)[/tex]:
Taking the positive square root (assuming [tex]\(\cos \theta\)[/tex] is positive):
[tex]\[
\cos \theta = \frac{|x - 1|}{x + 1}
\][/tex]
6. Calculate [tex]\(\tan \theta\)[/tex]:
Using the definition of [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2 \sqrt{x}}{x + 1}}{\frac{|x - 1|}{x + 1}} = \frac{2 \sqrt{x}}{|x - 1|}
\][/tex]
Now, let's consider an example to verify the calculations with a specific value for [tex]\(x\)[/tex]. Let [tex]\(x = 4\)[/tex]:
- [tex]\(\sin \theta = \frac{2 \sqrt{4}}{4 + 1} = \frac{2 \cdot 2}{5} = \frac{4}{5} = 0.8\)[/tex]
- [tex]\(\cos \theta = \frac{|4 - 1|}{4 + 1} = \frac{3}{5} = 0.6\)[/tex]
- [tex]\(\tan \theta = \frac{0.8}{0.6} \approx 1.3333\)[/tex]
Therefore, for [tex]\(x = 4\)[/tex], we have:
- [tex]\(\sin \theta = 0.8\)[/tex]
- [tex]\(\cos \theta = 0.6\)[/tex]
- [tex]\(\tan \theta \approx 1.3333\)[/tex]
Thus, the value of [tex]\(\tan \theta\)[/tex] is approximately 1.3333 when [tex]\(x = 4\)[/tex].
