The angles of a triangle are in AP and the greatest is double the least. Find all the angles in degrees and radians.

Explanation

To solve the problem of finding the angles of a triangle that are in arithmetic progression (AP) and where the greatest angle is double the least angle, we can follow these steps:

### Step 1: Define the angles
Let the least angle be \( x \) degrees. Since the angles are in AP, we can express the other two angles in terms of \( x \):
- The second angle (middle angle) can be denoted as \( y \).
- The greatest angle, which is double the least angle, will be \( 2x \).

### Step 2: Express the angles in terms of \( x \)
Since the angles are in AP, we can express them as:
- Least angle: \( x \)
- Middle angle: \( y \)
- Greatest angle: \( 2x \)

### Step 3: Use the property of AP
In an arithmetic progression, the difference between consecutive terms is constant. Therefore, we can set up the equation:
\[ y - x = 2x - y \]
This simplifies to:
\[ 2y = 3x \]
Thus, we can express \( y \) as:
\[ y = \frac{3x}{2} \]

### Step 4: Use the triangle angle sum property
The sum of the angles in a triangle is \( 180^\circ \). Therefore, we can write:
\[ x + y + 2x = 180^\circ \]
Substituting \( y \) from the previous step:
\[ x + \frac{3x}{2} + 2x = 180^\circ \]

### Step 5: Combine like terms
Combining the terms gives:
\[ x + 2x + \frac{3x}{2} = 180^\circ \]
This can be rewritten as:
\[ 3x + \frac{3x}{2} = 180^\circ \]
To combine these, convert \( 3x \) into a fraction:
\[ \frac{6x}{2} + \frac{3x}{2} = 180^\circ \]
Thus:
\[ \frac{9x}{2} = 180^\circ \]

### Step 6: Solve for \( x \)
Now, multiply both sides by 2:
\[ 9x = 360^\circ \]
Now divide by 9:
\[ x = 40^\circ \]

### Step 7: Find the other angles
Now that we have \( x \):
- The middle angle \( y \) is:
\[ y = \frac{3x}{2} = \frac{3 \times 40^\circ}{2} = 60^\circ \]
- The greatest angle is:
\[ 2x = 2 \times 40^\circ = 80^\circ \]

### Step 8: Summary of angles
Thus, the angles of the triangle are:
- Least angle: \( 40^\circ \)
- Middle angle: \( 60^\circ \)
- Greatest angle: \( 80^\circ \)

### Step 9: Convert to radians
To convert these angles to radians:
- \( 40^\circ = \frac{40 \pi}{180} = \frac{2\pi}{9} \) radians
- \( 60^\circ = \frac{60 \pi}{180} = \frac{\pi}{3} \) radians
- \( 80^\circ = \frac{80 \pi}{180} = \frac{4\pi}{9} \) radians

### Final Answer:
The angles of the triangle in degrees are \( 40^\circ, 60^\circ, 80^\circ \) and in radians are \( \frac{2\pi}{9}, \frac{\pi}{3}, \frac{4\pi}{9} \).

---