To find [tex]\(\tan \left(\frac{u}{2}\right)\)[/tex] given [tex]\(\cot u = 2.032\)[/tex] and [tex]\(u\)[/tex] is in the first quadrant, we can follow these steps:
1. Convert cotangent to tangent:
[tex]\[
\tan u = \frac{1}{\cot u}
\][/tex]
Given [tex]\(\cot u = 2.032\)[/tex], we get:
[tex]\[
\tan u = \frac{1}{2.032} \approx 0.4922
\][/tex]
2. Find sine and cosine of [tex]\(u\)[/tex]:
From [tex]\(\tan u = \frac{\sin u}{\cos u}\)[/tex], we use the identity:
[tex]\[
\sin^2 u + \cos^2 u = 1
\][/tex]
We can express [tex]\(\sin u\)[/tex] and [tex]\(\cos u\)[/tex] as:
[tex]\[
\sin u = \frac{\tan u}{\sqrt{1 + \tan^2 u}}
\][/tex]
[tex]\[
\cos u = \frac{1}{\sqrt{1 + \tan^2 u}}
\][/tex]
Using [tex]\(\tan u \approx 0.4922\)[/tex]:
[tex]\[
\sin u = \frac{0.4922}{\sqrt{1 + (0.4922)^2}} \approx 0.4421
\][/tex]
[tex]\[
\cos u = \frac{1}{\sqrt{1 + (0.4922)^2}} \approx 0.8969
\][/tex]
3. Use the half-angle formula for tangent:
[tex]\[
\tan \left(\frac{u}{2}\right) = \sqrt{\frac{1 - \cos u}{1 + \cos u}}
\][/tex]
Then substituting the values we found:
[tex]\[
\tan \left(\frac{u}{2}\right) = \sqrt{\frac{1 - 0.8969}{1 + 0.8969}} \approx \sqrt{\frac{0.1031}{1.8969}} \approx 0.2327
\][/tex]
Thus,
[tex]\[
\tan \left(\frac{u}{2}\right) \approx 0.2327
\][/tex]
Your answer is [tex]\(\boxed{0.2327}\)[/tex].
