The difference between two acute angles of a right angle triangle `(3pi)/10` rad. Find the angles in degree.

Answer

To solve the problem, we need to find two acute angles \( x \) and \( y \) of a right triangle, given that their difference is \( \frac{3\pi}{10} \) radians.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We know that in a right triangle, one angle is \( 90^\circ \) (or \( \frac{\pi}{2} \) radians). The other two angles, \( x \) and \( y \), must be acute angles. We are given that:
\[ x - y = \frac{3\pi}{10} \quad \text{(Equation 1)} \]

2. **Using the Angle Sum Property:**
The sum of the angles in a triangle is \( 180^\circ \). Thus, we can write:
\[ x + y + 90^\circ = 180^\circ \]
Simplifying this gives:
\[ x + y = 90^\circ \quad \text{(Equation 2)} \]

3. **Converting Degrees to Radians:**
To work with radians, we convert \( 90^\circ \) to radians:
\[ 90^\circ = \frac{\pi}{2} \quad \text{(Equation 2 in radians)} \]
So, we can rewrite Equation 2 as:
\[ x + y = \frac{\pi}{2} \]

4. **Solving the Equations:**
Now we have two equations:
- \( x - y = \frac{3\pi}{10} \) (Equation 1)
- \( x + y = \frac{\pi}{2} \) (Equation 2)

We can add these two equations to eliminate \( y \):
\[ (x - y) + (x + y) = \frac{3\pi}{10} + \frac{\pi}{2} \]
This simplifies to:
\[ 2x = \frac{3\pi}{10} + \frac{5\pi}{10} = \frac{8\pi}{10} \]
Therefore:
\[ x = \frac{8\pi}{20} = \frac{2\pi}{5} \]

5. **Finding \( y \):**
Now we can substitute \( x \) back into Equation 2 to find \( y \):
\[ \frac{2\pi}{5} + y = \frac{\pi}{2} \]
Rearranging gives:
\[ y = \frac{\pi}{2} - \frac{2\pi}{5} \]
To subtract these fractions, we need a common denominator, which is \( 10 \):
\[ y = \frac{5\pi}{10} - \frac{4\pi}{10} = \frac{\pi}{10} \]

6. **Converting Radians to Degrees:**
Now we convert \( x \) and \( y \) back to degrees:
- For \( x \):
\[ x = \frac{2\pi}{5} \times \frac{180}{\pi} = \frac{2 \times 180}{5} = 72^\circ \]
- For \( y \):
\[ y = \frac{\pi}{10} \times \frac{180}{\pi} = \frac{180}{10} = 18^\circ \]

### Final Answer:
The two acute angles are:
\[ x = 72^\circ, \quad y = 18^\circ \]