Certainly! Let's solve for [tex]\( u \)[/tex] and [tex]\( v \)[/tex] in a 30-60-90 triangle. A 30-60-90 triangle has special properties with specific ratios between the sides:
1. The side opposite the 30-degree angle is the shortest and we'll denote its length as [tex]\( a \)[/tex].
2. The side opposite the 60-degree angle is [tex]\( a\sqrt{3} \)[/tex].
3. The hypotenuse, opposite the 90-degree angle, is [tex]\( 2a \)[/tex].
Given this information, we need to find expressions for [tex]\( u \)[/tex] and [tex]\( v \)[/tex] using these ratios.
### Step-by-Step Solution:
1. Identify the lengths using the ratio:
- Let [tex]\( a \)[/tex] be the length of the side opposite the 30-degree angle.
- Then, the length of the side opposite the 60-degree angle is [tex]\( a\sqrt{3} \)[/tex].
- The length of the hypotenuse is [tex]\( 2a \)[/tex].
2. Define [tex]\( u \)[/tex] and [tex]\( v \)[/tex] based on the triangle:
- Let [tex]\( u \)[/tex] be the length of the side opposite the 60-degree angle.
- Let [tex]\( v \)[/tex] be the length of the hypotenuse.
3. Substitute the appropriate expressions into [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
- [tex]\( u = a\sqrt{3} \)[/tex]
- [tex]\( v = 2a \)[/tex]
### Conclusion:
Thus, the values of [tex]\( u \)[/tex] and [tex]\( v \)[/tex] in simplest radical form are:
[tex]\[
\begin{array}{l}
u = a\sqrt{3} \\
v = 2a
\end{array}
\][/tex]
These expressions follow directly from the known ratios in a 30-60-90 triangle.
