Certainly! Let's tackle this question step-by-step, making use of the given information.
### Step 1: Calculate cos(x)
Given:
[tex]\[
\sin(x) = \frac{3}{5}
\][/tex]
First, understand that for any angle [tex]\( x \)[/tex]:
[tex]\[
\sin^2(x) + \cos^2(x) = 1
\][/tex]
So, we can find [tex]\(\cos(x)\)[/tex] by rearranging this equation. Since [tex]\( x \)[/tex] is in the first quadrant, [tex]\(\cos(x)\)[/tex] is positive:
[tex]\[
\cos^2(x) = 1 - \sin^2(x)
\][/tex]
[tex]\[
\cos^2(x) = 1 - \left(\frac{3}{5}\right)^2
\][/tex]
[tex]\[
\cos^2(x) = 1 - \frac{9}{25}
\][/tex]
[tex]\[
\cos^2(x) = \frac{25}{25} - \frac{9}{25}
\][/tex]
[tex]\[
\cos^2(x) = \frac{16}{25}
\][/tex]
Thus:
[tex]\[
\cos(x) = \sqrt{\frac{16}{25}} = \frac{4}{5}
\][/tex]
### Step 2: Calculate tan(x)
Next, we find [tex]\(\tan(x)\)[/tex] using [tex]\(\sin(x)\)[/tex] and [tex]\(\cos(x)\)[/tex]:
[tex]\[
\tan(x) = \frac{\sin(x)}{\cos(x)}
\][/tex]
[tex]\[
\tan(x) = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}
\][/tex]
### Step 3: Calculate tan(x/2) using the half-angle identity
For the tangent of a half-angle, we use the half-angle identity:
[tex]\[
\tan\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}}
\][/tex]
Since [tex]\( x \)[/tex] is in the first quadrant, [tex]\( \tan\left(\frac{x}{2}\right) \)[/tex] will be positive:
[tex]\[
\tan\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \frac{4}{5}}{1 + \frac{4}{5}}}
\][/tex]
[tex]\[
\tan\left(\frac{x}{2}\right) = \sqrt{\frac{\frac{1}{5}}{\frac{9}{5}}}
\][/tex]
[tex]\[
\tan\left(\frac{x}{2}\right) = \sqrt{\frac{1}{9}} = \frac{1}{3}
\][/tex]
Thus, the value of [tex]\(\tan\left(\frac{x}{2}\right)\)[/tex] is:
[tex]\[
\tan\left(\frac{x}{2}\right) = \frac{1}{3}
\][/tex]
### Result
[tex]\[
\tan\left(\frac{x}{2}\right) \approx 0.33333333333333326
\][/tex]
This detailed step-by-step solution finds that the tangent of the half-angle, [tex]\(\tan\left(\frac{x}{2}\right)\)[/tex], is approximately [tex]\(0.33333333333333326\)[/tex].
